# 物理代写|高能物理代写High Energy Physics代考|FYS9550

## 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

$$F=\sigma T^4$$

## 物理代写|高能物理代写高能物理代考|恒星结构物理描述

，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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