“2018年IOAA理论第7题-太阳黑子”的版本间的差异
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(a)The pressure is from all directions. The plasma inside the sunspot must have the same total pressure as the plasma outside to maintain equilibrium on the border. | (a)The pressure is from all directions. The plasma inside the sunspot must have the same total pressure as the plasma outside to maintain equilibrium on the border. | ||
The kinetic pressure inside the sunspot is <br/> | The kinetic pressure inside the sunspot is <br/> | ||
| − | + | $$p_{i}=n_{i} k_{B} T_{i}$$<br/> | |
where n<sub>i</sub> is the number density inside the sunspot. <br/> | where n<sub>i</sub> is the number density inside the sunspot. <br/> | ||
And also,<br/> | And also,<br/> | ||
$$p_{e}=n_{e} k_{B} T_{e}$$<br/> | $$p_{e}=n_{e} k_{B} T_{e}$$<br/> | ||
| − | ( | + | (<math>p_{e} V=N_{e} k_{B} T_{e}</math> is also correct. But it’s better to write the pressure in terms of number density)<br/> |
From the assumption,n<sub>i</sub>=n<sub>e</sub>=n. The equilibrium requires:<br/> | From the assumption,n<sub>i</sub>=n<sub>e</sub>=n. The equilibrium requires:<br/> | ||
$$n k_{B} T_{i}+\frac{B_{i}^{2}}{2 \mu_{0}}=n k_{B} T_{e}+\frac{B_{e}^{2}}{2 \mu_{0}}$$<br/> | $$n k_{B} T_{i}+\frac{B_{i}^{2}}{2 \mu_{0}}=n k_{B} T_{e}+\frac{B_{e}^{2}}{2 \mu_{0}}$$<br/> | ||
2022年4月4日 (一) 11:12的版本
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英文原题
(T7) Sunspot (25 points)
Magnetic fields are important in the physics of stars and sunspots. As an approximation, we can model the photosphere of the Sun consisting of a plasma, which can be simply treated as a single component ideal gas, and a magnetic field (B), which has an associated magnetic pressure $$p_B=\frac{B^2}{2\mu_0}$$. It behaves like any other physical pressure except that it is carried by the magnetic field rather than by the kinetic energy of particles.
Assume that the number density of particles in the photosphere is constant everywhere, but the magnetic field inside the sunspot (Bin=0.1T) is much stronger than outside (Bout=5×10-3T). From the blackbody spectrum, the temperature inside the sunspot is Tin~4000K, while the temperature outside is Tout~6000K (which is why the sunspot looks darker). For the sunspot to be stable, the inside must be in equilibrium with the outside.
(a) Estimate the number density of plasma particles in the solar photosphere.
(b) Compare your answer with an estimate of the number density of particles in the atmosphere at the surface of the Earth.
中文翻译
在恒星和黑子的物理过程中磁场很重要。作为近似，我们将太阳光球层简化为一种等离子体，可以按照理想气体一样对待，并且具有一定的磁压$$p_B=\frac{B^2}{2\mu_0}$$，与当地的磁场（B）有关。磁压的行为与其他的压强类似，只不过它来自磁场而不是粒子的动能。
假设光球层中粒子的数密度是处处相等的，但是太阳黑子内部的磁场(Bin=0.1T)远大于外部(Bout=5×10-3T)。从黑体谱来看，太阳黑子内部的磁场为Tin~4000K，而外部的温度为Tout~6000K（这也就是为什么黑子看起来更暗）。为了使太阳黑子保持稳定，内部与外部的压强需要保持平衡。
(a)估算太阳光球层中粒子的数密度。
(b)将计算的结果与地表大气的数密度作比较。
英文解答
(a)The pressure is from all directions. The plasma inside the sunspot must have the same total pressure as the plasma outside to maintain equilibrium on the border.
The kinetic pressure inside the sunspot is
$$p_{i}=n_{i} k_{B} T_{i}$$
where ni is the number density inside the sunspot.
And also,
$$p_{e}=n_{e} k_{B} T_{e}$$
(\(p_{e} V=N_{e} k_{B} T_{e}\) is also correct. But it’s better to write the pressure in terms of number density)
From the assumption,ni=ne=n. The equilibrium requires:
$$n k_{B} T_{i}+\frac{B_{i}^{2}}{2 \mu_{0}}=n k_{B} T_{e}+\frac{B_{e}^{2}}{2 \mu_{0}}$$
Then
$$n=\frac{1}{2 \mu_{0} k_{B}} \frac{\left(B_{i}^{2}-B_{e}^{2}\right)}{\left(T_{e}-T_{i}\right)}=1.43 \times 10^{23} \mathrm{~m}^{-3}$$
(b)On the Earth, $p_{E}=n_{E} k_{B} T_{E}$ where $P_{E} \approx 10^{5} \mathrm{~Pa}$ and $T_{E} \approx 300 \mathrm{~K}$
Thus, $n_{E}=\frac{p_{E}}{k_{B} T_{E}}=2.4 \times 10^{25} m^{-3}$
This means that the number density of the atmospheric particles at the surface of the earth is at least 100 times larger that the number density of particles in the solar photosphere (of course the mass density is even higher).
