## 物理代写|天体物理学和天文学代写Astrophysics and Astronomy代考|Physics2151

2022年12月28日

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• Statistical Inference 统计推断
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• Foundations of Data Science 数据科学基础
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## 物理代写|天体物理学和天文学代写Astrophysics and Astronomy代考|Thermal Excitation and Ionization

Collisions between atoms excite some of them into a higher energy state, while others lose energy. The halance hetween these processes is described by the Boltzmann distribution. If the gas is in thermal equilibrium, the ratio of the occupation numbers $N_2$ and $N_1$ of levels $n_2$ and $n_1$, respectively, is given by
$$\frac{N_2}{N_1}=\frac{g_2}{g_1} \mathrm{e}^{-\left(E_2-E_1\right) / k T}$$
where $g_1$ and $g_2$ are the statistical weights of the energy levels (i.e. the number of possible quantum states with the same energy) and $T$ is the temperature of the gas. For the hydrogen atom, the $n$th energy level is degenerate with weight $g_n=2 n^2$ (the energy of a state is independent of the spin and the angular momentum quantum numbers).

As an example, let us compute $N_2 / N_1$ for the first excited state of hydrogen for the stars defined in the dictionary stars in Sect. 3.1.1:

The occupation numbers of the first excited state $\left(n_2=2\right)$ relative to the ground state $\left(n_1=1\right.$ ) are printed for the effective temperatures in the array T_sample (see Sect. 3.1.2). The Boltzmann equation (3.10) is implemented inline as an expression in terms of the variables $\mathrm{n} 1, \mathrm{n} 2$, and Teff. Line 5 defines the ionization energy $\chi \equiv E_{\infty}-E_1$ in terms of the Rydberg constant $R$ :
$$\chi=h c R=13.6 \mathrm{eV} .$$
This allows us to express the energy difference $\Delta E$ between the two states in Boltzmann’s equation as
$$\Delta E=-\chi\left(\frac{1}{n_2}-\frac{1}{n_1}\right) .$$
The value of $\chi$ in SI units (J) is available in SciPy’s physical_constants dictionary, which is imported in line 2 . This dictionary allows us to conveniently reference physical constants via keywords. In the case of $\chi$, the key expresses formula (3.11) in words. Since each item in physical constants is a tuple containing the numerical value, unit, and precision of a constant, we need to assign the first element of the tuple to the variable chi.

## 物理代写|天体物理学和天文学代写Astrophysics and Astronomy代考|The Balmer Jump

Balmer lines result from absorbed photons that lift electrons from the first excited state of hydrogen, $n_1=2$, to higher energy levels, $n_2>2$. If a photon is sufficiently energetic, it can even ionize a hydrogen atom. The condition for ionization from the state $n_1=2$ is
$$\frac{h c}{\lambda} \geq \chi_2=\frac{13.6 \mathrm{eV}}{2^2}=3.40 \mathrm{eV}$$
The corresponding maximal wavelength is $364.7 \mathrm{~nm}$. Like the higher Balmer lines, it is in the ultraviolet part of the spectrum. Since ionizing photons can have any energy above $\chi_2$, ionization will result in a drop in the radiative flux at wavelengths shorter than $364.7 \mathrm{~nm}$ rather than a line. This is called the Balmer jump.

To estimate the fraction of photons of sufficient energy to ionize hydrogen for a star of given effective temperature, let us assume that the incoming radiation is black body radiation. From the Planck spectrum (3.3), we can infer the flux below a given wavelength:
$$F_{\lambda \leq \lambda_0}=\pi \int_0^{\lambda_0} \frac{2 h c^2}{\lambda^5} \frac{1}{\exp (h c / \lambda k T)-1} \mathrm{~d} \lambda .$$
Since we know that the total radiative flux integrated over all wavelengths is given by Eq. (3.4), the fraction of photons with wavelength $\lambda \leq \lambda_0$ is given by $F_{\lambda \leq \lambda_0} / F$.
Since the integral in Eq. (3.17) cannot be solved analytically, we apply numerical integration. ${ }^{20}$ The definite integral of a function $f(x)$ is the area below the graph of the function for a given interval $x \in[a, b]$. The simplest method of numerical integration is directly based on the notion of the Riemann integral:
$$\int_a^b f(x) \mathrm{d} x=\lim {N \rightarrow \infty} \sum{n=1}^N f\left(x_{n-1 / 2}\right) \Delta x,$$
where $\Delta x=(b-a) / N$ is the width of the $n$th subinterval and $x_{n-1 / 2}=a+(n-$ $1 / 2) \Delta x$ is its midpoint. The sum on the right-hand side means that the area is approximated by $N$ rectangles of height $f\left(x_n\right)$ and constant width $\Delta x$. If the function meets the basic requirements of Riemann integration (roughly speaking, if it has no poles and does not oscillate within arbitrarily small intervals), the sum converges to the exact solution in the limit $N \rightarrow \infty$. In principle, approximations of arbitrarily high precision can be obtained by using a sufficient number $N$ of rectangles. This is called rectangle or midpoint rule.

# 天体物理学和天文学代考

## 物理代写|天体物理学和天文学代写Astrophysics and Astronomy代考|Thermal Excitation and Ionization

$$\frac{N_2}{N_1}=\frac{g_2}{g_1} \mathrm{e}^{-\left(E_2-E_1\right) / k T}$$

$$\chi=h c R=13.6 \mathrm{eV}$$

$$\Delta E=-\chi\left(\frac{1}{n_2}-\frac{1}{n_1}\right) .$$

## 物理代写|天体物理学和天文学代写Astrophysics and Astronomy代考|The Balmer Jump

$n_2>2$. 如果光子的能量足够大，它甚至可以电离氢原 子。状态电离的条件 $n_1=2$ 是
$$\frac{h c}{\lambda} \geq \chi_2=\frac{13.6 \mathrm{eV}}{2^2}=3.40 \mathrm{eV}$$

$$F_{\lambda \leq \lambda_0}=\pi \int_0^{\lambda_0} \frac{2 h c^2}{\lambda^5} \frac{1}{\exp (h c / \lambda k T)-1} \mathrm{~d} \lambda$$

$$\int_a^b f(x) \mathrm{d} x=\lim N \rightarrow \infty \sum n=1^N f\left(x_{n-1 / 2}\right) \Delta x$$

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## 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。