分段密度分布函数与太阳解析解模型的研究

发布时间：2018-11-26 06:52

【摘要】：太阳处在主序星阶段，其演化速率很慢，因而可以假定是一个球对称的流体静力学平衡系统，太阳内部的物理量只是径向距离的函数与角变量无关。在此基础上可以引入描述太阳内部结构的四个微分方程，即质量分布方程、流体静力学平衡方程、能量输运方程和能量平衡方程。研究太阳模型的主要任务就是在一定的边界条件下求解太阳的结构方程。随着计算机技术的发展，人们可以有效地求出结构方程的数值解，但是仍然存在一些关于太阳的未解之谜是数值解无法回答的。为了弥补数值解模型的不足，人们提出了许多太阳的解析解模型，并取得了许多有意义的结果。然而现有的解析解模型存在的突出问题是可调参数太少，模型不够灵活，计算结果与数值解模型的计算结果相差太大。此外，有些解析解模型（例如Stein模型）直接假设了密度分布函数的形式，却不能解释为什么要选取这种形式的密度分布函数。为此，本文提出了一个新的太阳解析解模型，根据物理性质的不同，把太阳内部划分成两个不同的区域，内部是核反应区，外部为非核反应区。首先考虑到物理学规律的要求，假设了密度分布函数所必须满足的特点，，并引入参数λ表示太阳中心密度与区域分界面处的密度的相对差别，引入参数α表示核反应区半径与太阳半径的比值。把核反应区与非核反应区的密度分布函数按照泰勒级数展开，取展开式的前几项，然后根据密度分布函数必须满足的特点，可以确定泰勒展开式的系数，从而很自然地引入了密度分布函数的具体表达式。新模型的实质是用二次函数模拟核反应区的密度分布函数，用一次函数模拟非核反应区的密度分布函数。如果已知太阳的密度分布函数和物态方程，可以不用考虑能量输运方程和能量平衡方程，直接求解质量分布方程和流体静力学平衡方程。通常把太阳看作理想气体，其物态方程满足理想气体状态方程。把太阳的密度分布函数依次代入质量分布方程、流体静力学平衡方程和理想气体状态方程，在特定的边界条件下，能够求解出太阳内部压强和温度的解析表达式。如果参数α和λ是已知的，就可以计算出太阳的中心密度、中心压强和中心温度的具体数值。本文最后讨论了可调参数对太阳中心处热力学参量的影响和新模型的优点。与其它解析解模型一样，新模型的优点是不用详细地考察太阳内部核反应的具体细节就能够给出太阳内部的热力学参数的解析表达式。通过对比，发现分段密度分布函数法太阳解析解模型比现有的解析解模型有了较大的改进，算出的太阳中心密度、中心压强和中心温度能够与解析解模型的计算结果相符合，并且能够极大地简化太阳发光度的计算，也能给出核反应区内能的解析表达式。由于新模型引入了两个可调参数，因而具有更大的灵活性，这对于我们研究太阳内部的物理过程很有意义。
[Abstract]:Due to the slow evolution rate of the Sun at the stage of the principal ordered star, it can be assumed that it is a spherically symmetric hydrostatic equilibrium system. The physical quantity of the sun is only a function of the radial distance independent of the angular variables. On this basis, four differential equations describing the inner structure of the sun, namely, mass distribution equation, hydrostatic equilibrium equation, energy transport equation and energy equilibrium equation, can be introduced. The main task of studying the solar model is to solve the structural equations of the sun under certain boundary conditions. With the development of computer technology, numerical solutions of structural equations can be obtained effectively, but there are still some unsolved riddles about the sun that cannot be answered by numerical solutions. In order to make up for the shortage of numerical solution models, many analytical models of the sun have been proposed, and many meaningful results have been obtained. However, the outstanding problem of the existing analytical solution model is that the adjustable parameters are too few, the model is not flexible enough, and the result of calculation is too different from that of the numerical solution model. In addition, some analytical solution models (such as Stein model) assume the form of density distribution function directly, but can not explain why the density distribution function is chosen in this form. In this paper, a new solar analytical solution model is proposed. According to the different physical properties, the solar interior is divided into two different regions. The inner part is a nuclear reaction zone, and the outer one is a non-nuclear reaction region. Firstly, considering the requirements of physical laws, the characteristics of density distribution function are assumed, and the parameter 位 is introduced to indicate the relative difference between the density of the solar center and the density at the regional interface. The ratio of the radius of the nuclear reaction zone to the radius of the sun is expressed by introducing the parameter 伪. The density distribution function of the nuclear reaction region and the non-nuclear reaction region is expanded according to Taylor series, the first few terms of the expansion are taken, and the coefficients of the Taylor expansion can be determined according to the characteristics that the density distribution function must satisfy. Thus the concrete expression of density distribution function is naturally introduced. The essence of the new model is to simulate the density distribution function of the nuclear reaction region by quadratic function and the density distribution function of the non-nuclear reaction region by the first order function. If the density distribution function and the equation of state of the sun are known, the mass distribution equation and the hydrostatic equilibrium equation can be solved without considering the energy transport equation and the energy balance equation. Generally, the sun is regarded as an ideal gas, and its equation of state satisfies the equation of state of an ideal gas. The density distribution function of the sun is substituted into the mass distribution equation, the hydrostatic equilibrium equation and the ideal gas state equation in turn. Under certain boundary conditions, the analytical expressions of the internal pressure and temperature of the sun can be obtained. If the parameters 伪 and 位 are known, the specific values of the central density, central pressure and central temperature of the sun can be calculated. Finally, the influence of adjustable parameters on the thermodynamic parameters at the center of the sun and the advantages of the new model are discussed. Like other analytical solution models, the advantage of the new model is that the analytical expression of the thermodynamic parameters of the solar interior can be obtained without detailed investigation of the specific details of the nuclear reaction in the solar interior. By comparison, it is found that the solar analytical solution model of the piecewise density distribution function method is much better than the existing analytic solution model. The calculated solar center density, central pressure and center temperature can agree with the results of the analytical solution model. The calculation of solar luminosity can be greatly simplified and the analytical expression of energy in nuclear reaction region can also be given. The new model has more flexibility because of the introduction of two adjustable parameters, which is of great significance for us to study the physical processes inside the sun.
【学位授予单位】：吉林大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：P182

【相似文献】