## 物理代写|天体物理学和天文学代写Astrophysics and Astronomy代考|UCASF3F5

2022年12月28日

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

The trajectory followed by an astronomical object is also called ephemeris, which derives from the Latin word for “diary”. In former times, astronomers observed planetary positions on a daily basis and noted the positions in tables. If you think about the modern numerical computation of an orbit, which will be discussed in some detail in the next chapter, you can imagine positions of a planet or star being chronicled for subsequent instants, just like notes in a diary.

In Sect.2.2, we considered the Keplerian motion of a planet around a star, which neglects the gravity of other planets. The orbits of the planets in the solar system can be treated reasonably well as Kepler ellipses, but there are long-term variations in their shape and orientation induced by the gravitational attraction of the other planets, especially Jupiter. To compute such secular variations over many orbital periods with high accuracy, analytical and numerical calculations using perturbation techniques are applied. An example is the VSOP (Variations Séculaires des Orbites Planétaires) theory [8]. The solution known as VSOP87 represents the time-dependent heliocentric coordinates $X, Y$, and $Z$ of the planets in the solar system (including Pluto) by series expansions, which are available as source code in several programming languages. We produced a NumPy-compatible transcription into Python.

The Python functions for the VSOP87 ephemerides are so lengthy that it would be awkward to copy and paste them into a notebook. A much more convenient option is to put the code into a user defined Python module. In its simplest manifestation, a module is just a file named after the module with the extension -py containing a collection of function definitions. The file vsop87.py is part of the zip archive accompanying this chapter. You can open it in any source code editor or IDE. After a header, you will find definitions of various functions and, as you scroll down, thousands of lines summing cosine functions of different arguments (since we use the cosine from numpy, we need to import this module in vsop87.py). As with other modules, all you need to do to use the functions in a Python script or in a notebook is to import the module:
| import vsop87 as vsop
However, this will only work if the file vsop87. py is located in the same directory as the script or notebook into which it is imported. If this is not the case, you can add the directory containing the file to the module search path. Python searches modules in predefined directories listed in sys . path You can easily check which directories are included by importing the sys module and printing sys . path. If you want to add a new directory, you need to append it to the environmental variable PYTHONPATH before starting your Python session. The syntax depends on your operating system and the shell you are using (search the web for pythonpath followed by the name of your operating system and you are likely to find instructions or some tutorial explaining how to proceed). Alternatively, you can always copy vsop87. py into the directory you are currently working.

## 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|Planetary Ephemerides

VSOP87 星历表的 Python 函数非常冗长，将它们复制并粘贴到笔记本中会很尴尬。一个更方便的选择是将代码放入用户定义的 Python 模块中。在其最简单的表现形式中，模块只是一个以模块命名的文件，扩展名为 -py，其中包含一组函数定义。文件 vsop87.py 是本章附带的 zip 存档的一部分。您可以在任何源代码编辑器或 IDE 中打开它。在标题之后，您会发现各种函数的定义，向下滚动时，您会发现数千行对不同参数的余弦函数求和（因为我们使用 numpy 的余弦函数，所以我们需要在 vsop87.py 中导入该模块）。与其他模块一样，要在 Python 脚本或笔记本中使用函数，您需要做的就是导入模块：
| 将 vsop87 导入为 vsop

## 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
$$\frac{h c}{\lambda} \geq \chi_2=\frac{13.6 \mathrm{eV}}{2^2}=3.40 \mathrm{eV}$$

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

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

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## MATLAB代写

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