{{需要解答}}
==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, 

$$p_e=n_e k_B T_e$$ 

(<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) 

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, <math>p_{E}=n_{E} k_{B} T_{E}</math> where <math>P_{E} \approx 10^{5} \mathrm{~Pa}</math> and <math>T_{E} \approx 300 \mathrm{~K}</math> 

Thus, <math>n_{E}=\frac{p_{E}}{k_{B} T_{E}}=2.4 \times 10^{25} m^{-3}</math> 

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).

==中文解答==
(a)压力是来自各个方向的。太阳黑子内外的等离子体的压强必须相等，使黑子边界处保持平衡。

太阳黑子内部的动力学压强为 

$$p_i=n_i k_B T_i$$ 

其中 ni 是黑子内的数密度 

同样有, 

$$p_e=n_e k_B T_e$$ 

(<math>p_e V=N_e k_B T_e</math> 也是正确的。但是最好写成压强和数密度的关系) 

由假设知,ni=ne=n。平衡要求: 

$$n k_{B} T_{i}+\frac{B_{i}^{2}}{2 \mu_{0}}=n k_{B} T_{e}+\frac{B_{e}^{2}}{2 \mu_{0}}$$ 
于是有 

$$n=\frac{1}{2 \mu_{0} k_{B}} \frac{\left(B_{i}^{2}-B_{e}^{2}\right)}{\left(T_{e}-T_{i}\right)}=1.43 \times 10^{23} \mathrm{~m}^{-3}$$ 

(b)在地球上， <math>p_{E}=n_{E} k_{B} T_{E}</math>，其中<math>P_{E} \approx 10^{5} \mathrm{~Pa}</math> ， <math>T_{E} \approx 300 \mathrm{~K}</math> 

于是有, <math>n_{E}=\frac{p_{E}}{k_{B} T_{E}}=2.4 \times 10^{25} m^{-3}</math> 

这意味着地球表面附近大气中粒子的数密度至少是太阳大气中等离子数密度的100倍（当然质量密度会更高）。
[[分类:磁场]]

2022-04-04T02:55:08Z

某九流辣鸡民科：/* 英文解答 */

2018年IOAA理论第7题-太阳黑子

2022-04-04T02:54:54Z

某九流辣鸡民科：

==英文题目==
A galaxy at the boundary of a galaxy cluster of radius 10 Mpc is expected to escape from the cluster if it has an initial velocity of at least 700 km/s relative to the center of the cluster. Calculate the density of the cluster.

2015IOAA理论短问题第6题-大气光深

2022-03-18T12:29:59Z

某九流辣鸡民科：创建页面，内容为“==英文题目== At the start of every observation, a radio telescope is pointed at a point-source calibrator that has a known flux density of 21.86 Jy outside the E…”

==英文题目==
At the start of every observation, a radio telescope is pointed at a point-source calibrator that has a known flux density of 21.86 Jy outside the Earth’s atmosphere. However, on a certain date, the measured flux density of the calibrator source was 14.27 Jy. If the calibrator source was at an altitude of 35 degrees, estimate the zenith atmospheric optical depth, $$\tau_{z}$$.

2015IOAA理论短问题第5题-卫星轨道

2022-03-18T12:29:04Z

某九流辣鸡民科：创建页面，内容为“==英文题目== An observer is trying to determine an approximate value of the orbital eccentricity of a man-made satellite. When the satellite was at apogee, it wa…”

==英文题目==
An observer is trying to determine an approximate value of the orbital eccentricity of a man-made satellite. When the satellite was at apogee, it was observed to have moved by $$∆𝜃_{1}$$ = 2′44" in a short time. When the radius vector connecting Earth and the satellite is perpendicular to the major axis (true anomaly is equal to 90o), within the same duration of time, it was observed to have moved by $$∆𝜃_{2}$$ = 21′17". Assume that the observer is located at the center of the Earth. Find an approximate value of the eccentricity of the satellite’s orbit.

2015IOAA理论短问题第4题-观测黑洞

2022-03-18T12:26:50Z

某九流辣鸡民科：创建页面，内容为“==英文题目== Let us assume that an observer using a hypothetical, far-infrared, Earth-sized telescope (wavelength range 20 to 640 $$\mu m$$) found a static and n…”

==英文题目==
Let us assume that an observer using a hypothetical, far-infrared, Earth-sized telescope (wavelength range 20 to 640 $$\mu m$$) found a static and neutral supermassive black hole with a mass of $$2.1 \times 10^{10} M_{\odot}$$. Determine the maximum distance at which this black hole can be resolved by the observer.

IOAA2015

2022-03-18T12:25:35Z

某九流辣鸡民科：/* 短问题 */

==理论考试==
====短问题====
PS:以下题目的标题为本页面编辑者自行命名

[[2015IOAA理论短问题第1题-共振轨道]]

[[2015IOAA理论短问题第2题-行星质量]]

[[2015IOAA理论短问题第3题-小行星掩星]]

[[2015IOAA理论短问题第4题-观测黑洞]]

[[2015IOAA理论短问题第5题-卫星轨道]]

[[2015IOAA理论短问题第6题-大气光深]]

[[2015IOAA理论短问题第7题-星团密度]]

[[2015IOAA理论短问题第8题-亮温度]]

==数据分析考试==

==观测考试==

==中国队参赛名单==

==参赛国家==

IOAA2015

2022-03-18T12:25:26Z

某九流辣鸡民科：/* 理论考试 */

==理论考试==
====短问题====
PS:以下题目的标题为本页面编辑者自行命名
[[2015IOAA理论短问题第1题-共振轨道]]

[[2015IOAA理论短问题第2题-行星质量]]

[[2015IOAA理论短问题第3题-小行星掩星]]

[[2015IOAA理论短问题第4题-观测黑洞]]

[[2015IOAA理论短问题第5题-卫星轨道]]

[[2015IOAA理论短问题第6题-大气光深]]

[[2015IOAA理论短问题第7题-星团密度]]

[[2015IOAA理论短问题第8题-亮温度]]

==数据分析考试==

==观测考试==

==中国队参赛名单==

==参赛国家==

2015IOAA理论短问题第3题-小行星掩星

2022-03-18T12:19:40Z

某九流辣鸡民科：创建页面，内容为“==英文题目== On 27 May 2015 at 02:18:49, the occultation of the star HIP 89931 (𝛿 −Sgr) by the asteroid 1285 Julietta was observed from Borobudur temple, wh…”

==英文题目==
On 27 May 2015 at 02:18:49, the occultation of the star HIP 89931 (𝛿 −Sgr) by the asteroid 1285 Julietta was observed from Borobudur temple, which was located at the center of the asteroid shadow path. It lasted for only 6.201 s. Assume that Earth’s orbit is circular and the orbit of Julietta is on the ecliptic plane and revolves in the same direction as Earth. At the occultation, Julietta is near its aphelion. At the time of the occultation the distances of Julietta from the Sun and the Earth are 3.076 AU and 2.156 AU respectively. Find the approximate diameter of asteroid Julietta, if the semi major axis of Julietta is 𝑎 = 2.9914 AU.

IOAA2015

2022-03-18T12:19:16Z

某九流辣鸡民科：/* 短问题 */

==理论考试==
====短问题====
[[2015IOAA理论短问题第1题-共振轨道]]

[[2015IOAA理论短问题第2题-行星质量]]

[[2015IOAA理论短问题第3题-小行星掩星]]

==数据分析考试==

==观测考试==

==中国队参赛名单==

==参赛国家==