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物理代写|高能物理代写High Energy Physics代考|Basic Facts and Observations
As already pointed out, the identification of lines in stellar spectra showed that stars were objects of the same type as the Sun. The presence of known chemical elements ( $\mathrm{H}, \mathrm{C}, \mathrm{O}$, etc.) and the study of spectral lines gives information about the outer stellar region, while its inner composition remains undetermined because the radiation does not carry information from the inner regions. But there are other ways to study stellar structure, at least indirectly. For example, it is relatively easy to determine how much energy is flowing from a star per unit frequency interval. In order to do this, it is enough to use filters that let through photons in some chosen range, and then count how many of them are in each wavelength interval. The total energy is easily calculated with the aid of the relation $E=h v$. In general, and for “normal” stars (the case of white dwarfs and others will be treated later), we find that there is a band where the number of photons reaches a maximum, and that the spectrum has approximately the shape of a black body spectrum. The black body, discussed by G. Kirchhoff and others at the beginning of the 20th century, is an idealization that applies to a perfect absorber/emitter which presents the distribution of intensities as a function of frequency shown in Fig. 4.2.
The colors of the stars correspond to the maximum of the distribution, since the photons with this wavelength are the most numerous, provided that the star complies with the black body idealization. The value of $\lambda_{\max }$ moves to lower wavelength values as the temperature increases. The lower the value of $\lambda_{\max }$ (i.e., the higher the frequency $v_{\max }$ ), the higher the temperature. Thus, the photons are said to be “harder” (i.e., more energetic) for distributions where the effective temperature is higher. If we restrict ourselves to the range marked $V$ (visible) in Fig. 4.2, the stars must present colors from red to blue, corresponding to temperatures between approximately 3800 and $10000 \mathrm{~K}$.
The next issue is the total energy emitted by the star, since we now have an idea of how the photons are distributed. The black body emission problem remained unsolved until the first years of the 20th century, as discussed in Chap. 2. In fact, the functional shape of the curves in Fig. $4.2$ corresponds to the expression (2.3), a consequence of the discrete nature of light. Note, however, that the total flux that emerges from a black body studied by G. Kirchhoff and others, i.e., the energy emitted per unit time and per unit area, has a very simple form: the result is proportional to the temperature to the fourth power, multiplied by a universal constant $\sigma$, and is completely independent of the composition of the body:
$$
F=\sigma T^4
$$
物理代写|高能物理代写High Energy Physics代考|Physical Description of Stellar Structure
The previous observation regarding the state of equilibrium and the fact that it can sustain stars for many millions of years leads to the question of the kind of equilibrium we are talking about. If we consider the case of a mass held up by a spring on the surface of the Earth (Fig. $4.4$ right), the mechanical balance of the system is guaranteed by the condition $\mathbf{F}{\text {grav }}=\mathbf{F}{\text {spring. But if we imagine now that the mass }}$ (in the form of a small cube) is an element of a fluid, the equivalent expression $\mathbf{F}{\text {grav }}=\mathbf{F}{\text {press }}$ points to two important considerations: first, the gravitational field is not “external” as in the case of the little cube on the Earth’s surface, but rather it is the very distribution of fluid that produces the gravitation, whence a star is often called a self-gravitating fluid; and second, the whole fluid distribution is also responsible for the force that sustains the “little cube” fluid element, by producing a pressure that balances the gravitation.
When dealing with a self-gravitating fluid, the mechanical balance of forces is called a hydrostatic balance. Free of any other forces, it is well known that the fluid will adopt a spherical form (to minimize its free energy). Thus, the hydrostatic equilibrium equation can be obtained by considering concentric shells of thickness $\mathrm{d} r$, where there is a pressure difference $P(r)-P(r+\mathrm{d} r)$ between the base and the top of any given shell. On the other hand, the shell is subject to the gravitational force that pulls it towards the center of the star (Fig. 4.5).
Now we can use calculus to express the forces in a simple way: the force produced by the pressure difference is $$
P(r)-P(r+\mathrm{d} r) \approx-\frac{\partial P}{\partial r} \mathrm{~d} r,
$$
and the gravitational pull is
$$
\mathbf{F}{\text {grav }}=-g(r) \rho(r) \mathrm{d} r=-\rho \frac{G M(r)}{r^2} \mathrm{~d} r, $$ where it is clear that the local acceleration due to gravitation increases as one moves outwards nwing to the accumulation of shells within, and should be calculated using the same density $\rho$ of the fluid. The condition $\mathbf{F}{\text {grav }}=\mathbf{F}_{\text {press }}$ leads immediately to
$$
\frac{\mathrm{d} P}{\mathrm{~d} r}=-\rho \frac{G M(r) \rho(r)}{r^2}
$$
物理代写|高能物理代写高能物理代考|基本事实和观察
正如已经指出的，恒星光谱中线条的识别表明，恒星是与太阳属于同一类型的物体。已知化学元素的存在($\mathrm{H}, \mathrm{C}, \mathrm{O}$等)和对光谱线的研究提供了关于恒星外层区域的信息，而其内部成分仍然不确定，因为辐射不携带来自内部区域的信息。但是还有其他的方法来研究恒星结构，至少是间接的。例如，相对容易确定每单位频率间隔有多少能量从一颗恒星流出。为了做到这一点，只需要使用过滤器，在选定的范围内让光子通过，然后计算每个波长区间内有多少光子。总能量很容易计算，借助于关系式$E=h v$。一般来说，对于“正常”恒星(白矮星和其他恒星的情况将在后面讨论)，我们发现存在一个光子数量达到最大值的波段，该光谱的形状近似于黑体光谱。G. Kirchhoff等人在20世纪初讨论的黑体，是适用于完美吸收体/发射体的一种理想化形式，它将强度分布表示为频率的函数，如图4.2所示
恒星的颜色对应于分布的最大值，因为这个波长的光子是最多的，前提是恒星符合黑体理想化。随着温度的升高，$\lambda_{\max }$的值向较低的波长值移动。$\lambda_{\max }$的值越低(即频率$v_{\max }$越高)，温度越高。因此，对于有效温度较高的分布，光子被称为“更硬”(即更有能量)。如果我们将自己限制在图4.2中标记为$V$(可见)的范围内，那么恒星的颜色必须从红色到蓝色，对应的温度大约在3800到$10000 \mathrm{~K}$之间
下一个问题是恒星发出的总能量，因为我们现在已经知道了光子是如何分布的。直到20世纪的头几年，黑体发射问题仍未解决，如第二章所述。事实上，图$4.2$中曲线的函数形状对应于表达式(2.3)，这是光的离散性质的结果。但是，请注意，G.基尔霍夫等人研究的黑体产生的总通量，即每单位时间和每单位面积发出的能量，有一个非常简单的形式:结果与温度的四次方成正比，乘以一个通用常数$\sigma$，完全与黑体的组成无关:
$$
F=\sigma T^4
$$
物理代写|高能物理代写高能物理代考|恒星结构物理描述
先前对平衡状态的观察，以及它可以维持恒星数百万年的事实，导致了我们正在讨论的那种平衡的问题。如果我们考虑一个物体被地球表面的弹簧支撑的情况(图右$4.4$)，系统的机械平衡由条件$\mathbf{F}{\text {grav }}=\mathbf{F}{\text {spring. But if we imagine now that the mass }}$(小立方体的形式)是流体的一个元素保证，等效表达式$\mathbf{F}{\text {grav }}=\mathbf{F}{\text {press }}$指出两个重要的考虑:首先，引力场不像地球表面的小立方体那样是“外部的”，而是流体的分布本身产生了引力，因此恒星常被称为自引力流体;第二，整个流体分布也负责维持“小立方体”流体元素的力，通过产生平衡重力的压力
当处理自引力流体时，力的机械平衡被称为流体静力平衡。在没有任何其他力的情况下，众所周知，流体将采用球形形式(以使其自由能最小化)。因此，流体静力平衡方程可以考虑厚度为$\mathrm{d} r$的同心壳，其中任何给定壳的底部和顶部之间有$P(r)-P(r+\mathrm{d} r)$的压差。另一方面，壳受到引力的作用，将其拉向恒星的中心(图4.5)
现在我们可以用微积分简单地表示这些力:压差产生的力是$$
P(r)-P(r+\mathrm{d} r) \approx-\frac{\partial P}{\partial r} \mathrm{~d} r,
$$
，重力拉力是
$$
\mathbf{F}{\text {grav }}=-g(r) \rho(r) \mathrm{d} r=-\rho \frac{G M(r)}{r^2} \mathrm{~d} r, $$，很明显，由于重力引起的局部加速度随着向外移动而增加，以内部堆积的贝壳为中心，应该使用相同的流体密度$\rho$来计算。条件$\mathbf{F}{\text {grav }}=\mathbf{F}_{\text {press }}$立即导致
$$
\frac{\mathrm{d} P}{\mathrm{~d} r}=-\rho \frac{G M(r) \rho(r)}{r^2}
$$
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