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# 物理代写|太阳系代写Solar System代考|PHYS7810 The Asteroid Belt

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## 物理代写|太阳系代写Solar System代考|The Asteroid Belt

The Asteroid Belt currently contains only enough material to make a planet 2000 times less massive than Earth, even though the spatial extent of the belt is huge. It seems likely that this region once contained much more mass than it does today. A smooth interpolation of the amount of solid material needed to form the inner planets and the gas giants would place about 2 Earth-masses in the Asteroid Belt. Even if most of this mass was lost at an early stage, the surface density of solid material must have been at least 100 times higher than it is today in order to grow bodies the size of Ceres and Vesta (roughly 900 and $500 \mathrm{~km}$ in diameter, respectively) in only a few million years.

Several regions of the Asteroid Belt contain clusters of asteroids with similar orbits and similar spectral features, suggesting they are made of the same material. These clusters are fragments from the collisional breakup of larger asteroids. There are relatively few of these asteroid families, which implies that catastrophic collisions are quite rare. This suggests the Asteroid Belt has contained relatively little mass for most of its history. The spectrum of asteroid Vesta, located 2.4 AU from the Sun, shows that it has a basaltic crust. The HED meteorites, which probably come from Vesta, show this crust formed only a few million years after the solar system, according to several isotopic systems. The survival of Vesta’s crust suggests that the crust formed the impact rate in the belt has never been much higher than it is today. For these reasons, it is thought that most of the Asteroid Belt’s original mass was removed at a very early stage by a dynamical process rather than by collisional erosion.
The Asteroid Belt currently contains a number of orbital resonances associated with the giant planets. Resonances occur when either the orbital period or precession period of an asteroid has a simple ratio with the corresponding period for one of the planets. Many resonances induce large changes in orbital eccentricity, causing asteroids to fall into the Sun, or to come close to Jupiter, leading to close encounters and ejection from the solar system. For this reason, there are very few asteroids that orbit the Sun twice every time Jupiter orbits the Sun once, for example. When the nebular gas was still present, small asteroids moving on eccentric orbits would have drifted inward rapidly due to gas drag. After the giant planets had formed, a combination of resonances and gas drag may have transferred most objects smaller than a few hundred kilometers from the Asteroid Belt into the terrestrial-planet region. Larger planetary embryos would not have drifted very far. However, once oligarchic growth ceased, embryos began to gravitationally scatter one another across the belt. Numerical simulations show that most or all of these bodies would eventually enter a resonance and be removed, leaving an Asteroid Belt greatly depleted in mass and containing no objects bigger than Ceres. The timescale for the depletion of the belt depends sensitively on the orbital eccentricities of the giant planets at the time, which are poorly known. The belt may have been cleared in only a few million years, but it may have required as much as several hundred million years if the giant planets had nearly circular orbits.

## 物理代写|太阳系代写Solar System代考|Growth of Gas and Ice Giant Planets

Jupiter and Saturn are mostly composed of hydrogen and helium. These elements do not condense at temperatures and pressures found in protoplanetary disks, so they must have been gravitationally captured from the gaseous component of the solar nebula. Observations of young stars indicate that protoplanetary disks survive for only a few million years, and this sets an upper limit for the amount of time required to form giant planets. Uranus and Neptune also contain significant amounts of hydrogen and helium (somewhere in the range 3-25\%), and so they probably also formed quickly, before the solar nebula dispersed.
Jupiter and Saturn also contain elements heavier than helium and they are enriched in these elements compared to the Sun. The gravitational field of Saturn strongly suggests it has a core of dense material at its center, containing roughly one fifth of the planet’s total mass. Jupiter may also have a dense core containing a few Earth masses of material. The interior structure of Jupiter remains quite uncertain because we lack adequate equations of state for the behavior of hydrogen at the very high pressures found in the planet’s interior. The upper atmospheres of both planets are enriched in elements such as carbon, nitrogen, sulfur, and argon, compared to the Sun. It is thought likely that these enrichments extend deep into the planets’ interiors, but this remains uncertain.

Giant planets may form directly by the contraction and collapse of gravitationally unstable regions of a protoplanetary disk. This disk instability is analogous to the gravitational instabilities that may have formed planetesimals, but instead the instability takes place in nebula gas rather than the solid component of the disk. Instabilities will occur if the Toomre stability criterion $Q$ becomes close to or lower than 1 , where
$$Q=\frac{M_{\mathrm{sun}} c_{\mathrm{s}}}{\sum \pi a^2 v_{\mathrm{kep}}}$$
where $v_{\mathrm{kep}}$ is the Keplerian velocity, $c_{\mathrm{s}}$ is the sound speed, and $\Sigma$ is the local surface density of gas in the disk. Gas in an unstable region quickly becomes much denser than the surrounding material. Disk instability requires high surface densities and low sound speeds (cold gas), so it is most likely to occur in the outer regions of a massive protoplanetary disk. Numerical calculations suggest instabilities will occur beyond about $5 \mathrm{AU}$ in a nebula a few times more massive than the minimum-mass solar nebula. What happens to an unstable region depends on how quickly the gas cools as it contracts, and this is the subject of much debate. If the gas remains hot, the dense regions will quickly become sheared out and destroyed by the differential rotation of the disk. If cooling is efficient, simulations show that gravitationally bound clumps will form in a few hundred years, and these may ultimately contract to form giant planets. Initially, such planets would be homogeneous and have the same composition as the nebula. Their structure and composition may change subsequently due to gravitational settling of heavier elements to the center and capture of rocky or icy bodies such as comets.

## 物理代写|太阳系代写太阳系代考|气体和冰巨行星的生长

$$Q=\frac{M_{\mathrm{sun}} c_{\mathrm{s}}}{\sum \pi a^2 v_{\mathrm{kep}}}$$
，其中$v_{\mathrm{kep}}$是开普勒速度，$c_{\mathrm{s}}$是声速，$\Sigma$是圆盘内气体的局部表面密度，则会发生不稳定。不稳定区域的气体密度很快就会比周围的物质大得多。盘的不稳定性需要较高的表面密度和较低的声速(冷气体)，所以它最有可能发生在一个大质量原行星盘的外部区域。数值计算表明，在质量比最小质量太阳星云大几倍的星云中，不稳定现象将在大约$5 \mathrm{AU}$以外的地方发生。不稳定区域会发生什么取决于气体收缩时的冷却速度，这是一个备受争议的话题。如果气体保持高温，密集区域将很快被剪切出来，并被圆盘的差动旋转破坏。如果冷却是有效的，模拟表明受引力束缚的团块将在几百年内形成，这些团块可能最终收缩形成巨大的行星。最初，这样的行星应该是均匀的，具有与星云相同的成分。它们的结构和组成可能随后发生变化，这是由于较重元素的引力沉降到中心，以及彗星等岩石或冰体的捕获

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。