村嶋慶哉「Modification of SPH for the three-dimensional simulation of an icy moon with internal ocean」(2021年度)

There are some traces of existence of internal ocean in some icy moons, such as a plumes of vapor of Europa and Enceladus. This suggests that there is a region of liquid water beneath the surface ice shell. Since liquid water would be essential for the origin of life, it is important to understand the development of internal ocean, especially the temperature distribution/evolution inside the icy moons. It is considered that the balance between the tidal heating caused by tidal acceleration and radiative cooling sustains liquid water beneath the surface of an icy moon. We aim to simulate tidal heating an internal ocean of an icy moon by 3-dimensional numerical fluid calculations using Smoothed Particle Hy- drodynamics (SPH) method. The SPH method is a particle-based method for simulations of fluid dynamics and commonly used for problems involving fracture or large deformation. We added viscosity and thermal conduction terms into the governing equations of SPH. However, we found two problems in calculating the rigid body rotation by using SPH including viscous term: (1) With the conven- tional formulation of viscosity, unwanted viscosity force prevent the rotation. (2) There is an artificial internal energy partitioning in the layered structure, which is due to the formulation of standard SPH. In order to resolve the former problem, we modified the formulation of viscosity and verified that unwanted viscous force is suppressed by it. For the latter problem, we introduced Density Independent SPH (DISPH), which was developed to improve the behavior of discontinuous surface. In addition, by using the algorithm to define the surface of a fluid using the particle method, we introduced cooling by radiation from the satellite surface. Also, we introduced an equation of state that takes phase transitions into account. According to the above modifications, we have developed the SPH method that introduces all the necessary physical processes (viscosity, conduction, radiative cooling and phase transition) to simulate the evolution of an internal ocean of an icy moon.

藤田菜穂「Orbital Evolution of Close-in Super-Earths via Photoevaporation」(2021年度)

Recently, over 4500 exoplanets have been discovered thanks to NASA’s Kepler space telescope. Exoplanet statistics show that low-mass planets with masses about 1–10 M⊕, which are called “super-Earths,” account for the majority of exoplanets and they have a variety of characteristics. Some of the super-Earths have a substantial atmosphere, while others have almost no atmosphere, according to their mass-radius relationships. In addition, the observed exoplanet population has some unique features. There is a scarcity of close-in small planets with 1.5– 2.0 R⊕ and orbital periods of < 100 days, which is called a “radius gap.” Also, there is a lack of close-in Neptune-sized planets with orbital periods of ≲ 5 days, which is called a “Neptune desert.” Although the origins of a variety of super-Earths and the observed features of the exoplanet population remain unclear, atmospheric escape from planets should play an important role. Indeed, previous studies showed that the radius gap can be reproduced by photoevaporation of super-Earths (e.g., Owen & Wu 2017). These studies considered the in-situ mass loss of a close-in planet via photoevaporation. Planets that undergo the mass loss, however, move outward due to the change in the orbital angular momentum of their star-planet systems. In this study, we calculated the orbital evolution of an evaporating super-Earth with a H2/He atmosphere around FGKM-type stars under a stellar X-ray and ex- treme UV (XUV) irradiation. Then, we investigated whether the observed features of the exoplanet population, such as the radius gap and the Neptune desert, appear in the period-radius distribution of close-in planets, even when we take into account the mass-loss-driven orbital evolution. As a result, we confirm that close-in super-Earths around FGKM-type stars should have experienced outward migration for ∼1Gyr if they had a primordial atmosphere. The rate of increase in the orbital radius of an evaporating planet is approximately proportional to that of the atmospheric mass loss during a high stellar XUV phase. The changes in the orbits of the planets would be small, but they can affect the orbital configurations of the compact, multi-planet systems, such as the TRAPPIST-1 system, whose orbital separations between each planet are ∼ 10% to their planetary orbital radius. Furthermore, we find that the radius gap and the Neptune desert in the observed population of close-in planets around FGK-type stars can be reproduced in our simulations; that is, our model, which includes the orbital evolution, is also consistent with observations. On the other hand, the observed planet population around M-type stars can be reproduced only by a high stellar XUV luminosity model. The detection of more super-Earths by ongoing planet surveys will enable us to verify our model, and then gain a better understanding of the atmospheric and orbital evolution of super-Earths.


かつて地球に火星サイズの天体が衝突し、その際に飛散した破片が再集積して月ができたとする巨大天体衝突説は、地球の月の形成過程として有力視されている。惑星形成過程の最終段階ではこのような巨大天体同士の衝突は頻繁に生じると考えられており、それに伴った衛星形成は普遍的に起こりうると考えられる。地球への巨大天体衝突に関して、smoothed particle hydrodynamics (SPH) 法を用いて流体シミュレーションがなされ、破片からなるデブリ円盤が地球の周囲に広がることが示唆された。さらに、デブリ円盤に含まれる破片が衝突による合体を繰り返して成長し、現在の月サイズになりうることが N 体計算で示された。このようにして、地球の月に対する巨大天体衝突説は数値計算的にも検証されてきた。一方で、地球の月以外においては巨大天体衝突による衛星系形成の研究が十分なされているとは言い難い。近年、大量の系外惑星が発見され、系外衛星候補も報告されていることから、系外衛星は今後重要な観測ターゲットになることが期待される。また、衛星の有無は惑星の居住可能性に影響を与えるため、一般的な惑星の周囲に衛星が形成する条件を調べることは地球外生命探査の観点からも重要である。そのため、巨大天体衝突に伴う衛星形成過程により、系外惑星を含めた一般的な惑星の周囲にどのような衛星系が形成されうるかを検証することは重要である。巨大天体衝突から衛星の形成までを一気通貫にシミュレーションすることは数値計算の都合上困難である。よって、一般的な惑星形成過程で生じる巨大天体衝突のパラメータを推定し、そのような衝突から生じるデブリ円盤の特性を推定し、そのようなデブリ円盤から形成する衛星系を推定する必要がある。そこで、本研究では最後の部分に焦点を当て、必ずしも地球を想定しない多様な質量・半径のデブリ円盤に対して N 体計算を行い、どのような衛星系ができるのかを検証した。初期円盤質量・初期円盤半径と形成する衛星の位置・個数・質量の関係を調べたところ、ロッシュ限界半径のやや外側に最大の衛星ができ、初期円盤半径が大きい場合にはさらにその外側に複数個の衛星ができることがわかった。また、最大の衛星の質量は初期円盤質量・初期円盤半径と関係性があることがわかった。

木原遥大「N-body simulations of Uranian satellites’ formation」(2021年度)

Uranus’s rotation axis is tilted by 98 degrees. In addition, Uranian satellite system is also tilted. Because of the above characteristics, Uranus is considered to have experi- enced a giant impact. Herein, based on the debris disk model derived by a previous work, we performed N-body simulations and confirmed whether we can reproduce the current Uranian satellite system. we found that in most cases, the smaller satellites were formed in the inner area and the larger ones in the outer area. It was consistent with the current Uranian satellites system. On the other hand, there were two points which were not consistent with observations. First, we were not able to reproduce Ariel. The reason was the power of the surface density of the initial disk was too steep (Σ ∝ r1.5), and the surface density was too small at the location where Ariel located. A satellite that was formed in the outer region could move into the Ariel’s location, however it would take too much calculation time to verify it. Second, multiple satel- lites were produced outside of Oberon. No satellites have been found in that area. In order to resolve the discrepancy, we considered a protoplanet transit after satellites formed, but to confirm it , we need to estimate the probability of an encounter between Uranus and a protoplanet.

角田伊織「N 体計算による準惑星ハウメアのリング形成過程の検証」(2019年度)

準惑星ハウメアは、リングを持つ唯一の太陽系外縁天体である。ハウメアのリングは 2017 年に恒星の掩蔽観測によって発見された。そのリングは、ハウメアの自転周期と 3:1 の平均運動共鳴を起こす位置にある (Ortiz et al. 2017)。ハウメアは三軸不等楕円体の形状をしており、その周囲の非軸対称重力場がリングの力学に影響を及ぼしていると考えられるが、リングの形成過程については解明されていない。我々は、ハウメアの自転による分裂によってハウメアの 2 つの衛星が形成されたという説 (Ortiz et al. 2012) に着目し、このモデルに基づいて以下のようなハウメア系形成のシナリオを提示した。まず、ハウメアから衛星サイズの破片が複数飛び散ったという状況を考える。ハウメア周囲の非軸対称重力場のため、ハウメアの近傍では物体が安定して存在できない。また、安定軌道にある物体のうち、ロッシュ半径の内側にあるものは、潮汐力によって破壊され、それがハウメアを公転することでリングになる。ロッシュ半径の外側に位置していた物体は、潮汐で軌道進化し、現在の衛星の位置まで移動する。以上のシナリオのうち、本研究では、軌道不安定領域の外側かつロッシュ半径の内側に位置している物体が潮汐破壊され、リングになる過程を検証する。まず、三軸不等楕円体の周囲の重力場を計算し、時間変動する重力場を組み込んだシミュレーションにより、ハウメアを公転する物体が安定して存在できる領域を見積もった。その結果、非常に短いタイムスケールで 2:1 軌道共鳴より内側の粒子が取り除かれ、最終的な不安定領域境界はリング半径の 0.905 倍の位置となった。すなわち、ちょうど現在のリングの位置よりも内側で、物体の軌道が不安定となることがわかった。次に、三軸不等楕円体周囲を公転する完全剛体に対するロッシュ半径を解析的に導出したところ、剛体ロッシュ半径はリング半径の 0.71 倍になった。また、ハウメアを同体積の球に近似したときの流体ロッシュ半径はリング半径の 1.09 倍になった。解析的な計算により、ロッシュ半径の位置が現在のリングの位置付近になる可 能性が示された。さらに、パラメータスタディとして、ラブルパイル物体の反発係数を変数とした N 体シミュレーションを行ったところ、多くのパラメータにおいて、ロッシュ半径の位置が現在のリングの位置付近になることが示された。以上のことから、軌道不安定領域境界とロッシュ半径の間の領域にリングが形成されるという本研究のシナリオが正しい可能性が高いことがわかった。よって、本研究で提示したシナリオによって、リング形成過程を説明できる可能性があることがわかった。

山中陽裕「Orbital Evolution of a Circumbinary Planet within Dynamically Unstable Region」(2018年度)

Among 3869 extrasolar planets known today, 21 circumbinary planets (CBPs), which orbit around two stars, have been discovered. Although CBPs take a small proportion of known exoplanets, it is important to study planet formation processes in binary systems because the formation processes of discovered CBPs are unclear and planetary systems around binary stars may have different charaster- istics than those around single stars. Moreover, about a half of solar type stars are considered to be multiple systems, suggesting more undiscovered CBPs may exist. Sub-Jupiter class CBPs discovered in close-in binary systems have orbits just beyond the dynamically unstable region determined by the eccentricity and mass ratio of the host binary stars. Since the planets’ semi-major axes fall within 1 au they are assumed to have formed in the outer area and migrated to the current orbit rather than forming in situ, but previous attempts to reproduce their orbit have not succeeded. We propose a scenario in which a planet formed in the outer region migrates to the inner edge of the circumbinary disk and then moves to the current orbit through outward migration. This migration is driven by the balance of orbital excitation of the central stars inside the gravitationally unstable region and damping by the gas drag force. We developed a N-body calculation code and carried out N-body simulations with a dissipating circumbinary protoplanetary disk for binary systems with different eccentricities and mass ratios. Results show that planets are more likely to achieve a stable orbit just beyond the unstable region in less eccentric binary systems. This result is not as sensitive to mass ratio as it is to eccentricity. These dependencies are consistent with the data from observed binary systems hosting CBPs. We derived rates of planets successfully achieving stable orbits close to the unstable boundary for each set of binary parameters. The planet-surviving parameter range covers most of observed CBP systems, suggesting our results support the idea that circumbinary planets’ orbits close to the unstable boundary result from outward migration from a gravitationally unstable orbit to the present orbit and shows probability of more CBPs discovered near unstable boundary by searching for transit signals with periods close to the predicted boundary for the system. in future observations.


月は地球をはじめとした他の太陽系天体と比較して特有の性質を持つため、どのようにして形成されたかという問題は極めて難しい問題である。現在の月形成は巨大衝突説と呼ばれるシナリオが最も有力である。巨大衝突説では形成初期の地球に火星サイズほどの原始惑星が衝突(Giant impact)し、地球の周りに気液共存状態の原始月円盤を形成すると考えられている。しかし、Giant impactからの円盤形成、そしてその円盤が十分にcoolingした後における月の形成については先行研究の数値計算により具体的なタイムスケールが算出されているが、その間の円盤のcoolingの状態は不明瞭な点が多く、様々な円盤モデルが乱立し、考えられるタイムスケールも幅広い値を取っている。また、このcoolingの過程が月形成の律速段階と考えられるので、月物質の化学進化にも大きく影響を与える。本研究ではcoolingのタイムスケールを評価するために、coolingについての先行研究の仮定について検証した。すなわち原始月円盤中の液滴にRayleigh-Taylor不安定性を考慮することで、Machida & Abe (2004)が提唱した原始月円盤モデル:Stratify modelとは異なるモデルを考えた。この場合、円盤進化の過程として2つのシナリオが考えられるが、そのどちらも拡散のタイムスケールは~10^2 yrになることが分かった。また、Sharnoz&Michaut(2015)が用いた3D-modeの分散関係の仮定である非圧縮性がStratify modelの液体層へ適用可能な条件について調べた。その結果、液体層のgasmassfractionと温度が液体層の非圧縮性を決定する重要なファクターであることを明らかにした。今後は、本研究で明らかになった原始月円盤中の液滴へのRayleigh-Taylor不安定性の影響と、Stratify modelの液体層の非圧縮性条件を考慮した上で、原始月円盤の粘性進化を考察していくことが重要である。

石澤祐弥「In-situ formation of the Uranian satellites from debris disk formed by giant impact」(2017年度)

Uranus has a 98◦ tilt of the rotational axis with respect to the plane of Solar System, whereas the regular satellites of Uranus orbit in the plane of its equatorial plane. Several scenarios have been proposed so far to explain the large tilt and the origin of the satellites respectively (e.g., Slattery et al., 1992; Canup and Ward, 2006; Crida and Charnoz, 2012). In this study, I adopted the so-called giant impact scenario, which could explain both the large tilt of Uranus and the formation of the regular satellites simultaneously. The hydrodynamic simulations of the giant impact have been carried out using the smoothed particle hydrodynamics (SPH) method (Slattery et al, 1992; Ueta et al., in prep.). They suggested that the giant impact of an Earth-sized protoplanet with proto-Uranus could tilt the rotational axis, and a circum-planetary debris disk would be produced throughout the current Uranian satellites orbits by the impact. However, it is still unknown whether the Uranian satellites can be actually formed from such a wide disk. Here I modeled a wide debris disk of solids with several conditions, performed N-body simulations to investigate the in-situ satellite formation from the debris disk and also discussed what kind of debris disks is suitable for the in-situ formation. I used a 4th order Hermite scheme and Leap frog method for the numerical integration, and considered the gravity, collision and merger between each particle (Kokubo et al., 2000). I found that satellites with the similar orbital radii and masses to the current satellite are formed in the outer region from 5RU to 25RU, where RU indicates the Uranian radius. Such satellites are formed under the conditions both when the power-index of the surface density distribution of the disk is larger than or equal to roughly -2 and when the initial disk mass is around 3 × 10^−4MU, which is corresponding to three times of the the satellite system mass, where MU indicates the Uranian mass. However, I also found that in the inner region 2.5RU to 5RU satellites are generally formed with much larger mass compared to the current satellites in the same region. I propose an additional scenario of orbital evolution to explain the inner satellite distribution as the following; After the in-situ formation in a wide circum-planetary disk, the inner satellites migrate inward onto Uranus due to the tidal torque of Uranus and the tidal dissipation inside the satellites, then the satellites falling into the inside of 2.5RU are disrupted by the planetary tides. The disrupted satellites can form rings around Uranus and small satellites are secondarily formed from the rings in the way proposed by Crida and Charnoz (2012). The outer satellites stay almost in their orbits since they are too far from Uranus to migrate. I speculate that thirteen inner satellites and five major satellites of Uranus are formed in different ways. I analytically calculated the orbital evolutions of the five major satellites from the past in several cases and found that the outer three satellites can stay almost in their orbits during 4.5 billion years even if under the situation of extremely strong tides of Uranus. Satellite’s orbit changes also due to the gravitational interaction with the disk in addition to the tides. It would play a key role to explain the satellite distribution and should be investigated in more detail in the future.


月は地球をはじめとした他の太陽系天体と比較して特有の性質を持つため、どのようにして形成されたかという問題は極めて難しい問題である。現在の月形成は巨大衝突説と呼ばれるシナリオが最も有力である。巨大衝突説では形成初期の地球に火星サイズほどの原始惑星が衝突し、その破片が再び自己重力によって集まって形成されて月が形成されたと考えられている。しかし現在でも揮発性元素 (約 1000 K 以下の温度で蒸発する元素) の枯渇などの月の組成の特徴については完全には説明できてはいない。本研究では円盤の鉛直方向の 1 次元モデルを用いて、円盤の光球面の温度を計算した。従来考えられていた温度は太陽系空間での岩石の凝縮温度である 2000 K であったが、今回の研究結果から円盤の光球面の温度は 500 K ほどまで小さくなることが明らかになった。そのため円盤の冷却時間はこれまでよりも十分長くなることが示唆され、最速のシナリオでも約 1 yr、最大のシナリオでは何と約 107 yr もの時間がかかることが分かった。これは円盤形成時間 (1 日) や月集積時間 (1 ヶ月) よりも十分長いため巨大衝突説で月形成のタイ ムスケールは原始月円盤が決定するものと思われる。また円盤中の乱流についても考察した。月の揮発性元素の枯渇の原因は円盤の乱流によって蒸気が地球に粘性降着したものであると想定した。今回求めた円盤の寿命と角運動量輸送時間を比較することで円盤の乱流の大きさに制限を与えた。その結果、円盤内の乱流は非常に穏やかな乱流であることが分かった。今後は、本研究で明らかになった原始月円盤の冷却時間および乱流の大きさを用いて、大規模な月形成の数値計算を行っていくことが重要である。

上田翔士「Development of Numerical Code based on the DISPH Method for Simulation of the Impacts of Small Planetary Bodies」(2013年度)

The evolution of atmosphere and ocean on the Earth is significantly influenced by the impact of small planetary bodies. The modified atmosphere and ocean could change the surface environment, and it may determine the habitability of the planet. While some physical mechanisms causing atmospheric erosion by impact have been investigated, a comprehensive understanding of the impact-induced atmospheric erosion process is lack- ing. The most realistic numerical simulations, Shuvalov (2009) and Shuvalov et al. (2013), assumed only rock material as the target of impacts, and did not consider the oceanic erosion. In this study, we aim to develop an advanced numerical code for simulation of impacts of small bodies, assuming the target as land and/or ocean. We use a new Lagrangian hydrocode in Hosono et al. (2013), Density Independent Smoothed Particles Hydrodynamics (DISPH). In the hydrostatic equilibrium tests, the contact discontinuity with quite large difference of density, such as the boundaries between the atmosphere and ocean/land, can be expressed exactly by using the DISPH method with unequal- mass particles and equal-separation arrangement. A numerical code for simulations of impacts is developed in this work with various impact parameters, such as projectile di- ameter, impact velocity, impact angle, projectile material, and target material. From the preliminary simulations of impacts with the numerical code, we find that the variation of impact parameters makes a large difference to the picture after the collision. We have some problems in employing realistic non-ideal EOS suitable for these simulation and setting the initial conditions. However, we find that we can express exactly the contact discontinuity with quite different values of density by using the DISPH code. This result is of great importance for calculation load as well, and it might also help us solve other unsettled problems in astrophysical and planetary sciences field.