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Astronomy_1160

In science fiction films the asteroid belt is typically portrayed as a region of the Solar System where the spacecraft needs to dodge and weave its way through many large asteroids that are rather close together. However, if this image were true then very few probes would be able to pass through the belt into the outer Solar System. [figure1] Figure 1 Artist conceptual illustration of the asteroid belt (left). Schematic of the Solar System with the asteroid belt between Mars and Jupiter (right). This question will look at the real distances between asteroids.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. problem: In science fiction films the asteroid belt is typically portrayed as a region of the Solar System where the spacecraft needs to dodge and weave its way through many large asteroids that are rather close together. However, if this image were true then very few probes would be able to pass through the belt into the outer Solar System. [figure1] Figure 1 Artist conceptual illustration of the asteroid belt (left). Schematic of the Solar System with the asteroid belt between Mars and Jupiter (right). This question will look at the real distances between asteroids. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of \mathrm{~W}, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

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Astronomy

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multi-modal

Astronomy_501

如图所示为人类历史上第一张黑洞照片。黑洞是一种密度极大、引力极大的天体, 以至于光都无法逃逸, 科学家一般通过观测绕黑洞运行的天体的运动规律间接研究黑洞。已知某黑洞的逃逸速度为 $v=\sqrt{\frac{2 G M}{R}}$, 其中引力常量为 $G, M$ 是该黑洞的质量, $R$ 是该黑洞的半径。若天文学家观测到与该黑洞相距为 $r$ 的天体以周期 $T$ 绕该黑洞做匀速圆周运动，则下列关于该黑洞的说法正确的是（） [图1] A: 该黑洞的质量为 $\frac{G T^{2}}{4 \pi r^{3}}$ B: 该黑洞的质量为 $\frac{4 \pi r^{3}}{G T^{2}}$ C: 该黑洞的最大半径为 $\frac{4 \pi^{2} r^{3}}{c^{2}}$ D: 该黑洞的最大半径为 $\frac{8 \pi^{2} r^{3}}{c^{2} T^{2}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 如图所示为人类历史上第一张黑洞照片。黑洞是一种密度极大、引力极大的天体, 以至于光都无法逃逸, 科学家一般通过观测绕黑洞运行的天体的运动规律间接研究黑洞。已知某黑洞的逃逸速度为 $v=\sqrt{\frac{2 G M}{R}}$, 其中引力常量为 $G, M$ 是该黑洞的质量, $R$ 是该黑洞的半径。若天文学家观测到与该黑洞相距为 $r$ 的天体以周期 $T$ 绕该黑洞做匀速圆周运动，则下列关于该黑洞的说法正确的是（） [图1] A: 该黑洞的质量为 $\frac{G T^{2}}{4 \pi r^{3}}$ B: 该黑洞的质量为 $\frac{4 \pi r^{3}}{G T^{2}}$ C: 该黑洞的最大半径为 $\frac{4 \pi^{2} r^{3}}{c^{2}}$ D: 该黑洞的最大半径为 $\frac{8 \pi^{2} r^{3}}{c^{2} T^{2}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

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multi-modal

Astronomy_305

如图所示, 假设地球半径为 $R$, 地球表面的重力加速度为 $g$, 飞船在轨道半径为 $R$的近地圆轨道I上运动, 到达轨道上 $A$ 点时点火进入椭圆轨道II, 到达粗圆轨道II的远地点 $B$ 点时再次点火进入距地而高度为 $3 R$ 的圆轨道III绕地球做圆周运动, 不考虑飞船质量的变化，点火时间极短。下列分析正确的是（） [图1] A: 飞船在轨道 I 上绕地球运行的周期为 $2 \pi \sqrt{\frac{R}{g}}$ B: 飞船在 II、III 轨道上通过 $B$ 点的加速度相等 C: 飞船在轨道 III 上运行速率为 $\sqrt{\frac{g R}{3}}$ D: 飞船在轨道 I 上的机械能比在轨道 III 上的机械能大

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 如图所示, 假设地球半径为 $R$, 地球表面的重力加速度为 $g$, 飞船在轨道半径为 $R$的近地圆轨道I上运动, 到达轨道上 $A$ 点时点火进入椭圆轨道II, 到达粗圆轨道II的远地点 $B$ 点时再次点火进入距地而高度为 $3 R$ 的圆轨道III绕地球做圆周运动, 不考虑飞船质量的变化，点火时间极短。下列分析正确的是（） [图1] A: 飞船在轨道 I 上绕地球运行的周期为 $2 \pi \sqrt{\frac{R}{g}}$ B: 飞船在 II、III 轨道上通过 $B$ 点的加速度相等 C: 飞船在轨道 III 上运行速率为 $\sqrt{\frac{g R}{3}}$ D: 飞船在轨道 I 上的机械能比在轨道 III 上的机械能大 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

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Astronomy

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multi-modal

Astronomy_283

如图所示, 某次发射远地圆轨道卫星时, 先让卫星进入一个近地的圆轨道I, 在此轨道运行的卫星的轨道半径为 $R_{1}$ 、周期为 $T_{1}$; 然后在 $P$ 点点火加速, 进入陏圆形转移轨道II, 在此轨道运行的卫星的周期为 $T_{2}$; 到达远地点 $\mathrm{Q}$ 时再次点火加速, 进入远地圆轨道III, 在此轨道运行的卫星的轨道半径为 $R_{3}$ 、周期为 $T_{3}$ (轨道II的近地点为 $\mathrm{I}$ 上的 $\mathrm{P}$ 点,远地点为轨道III上的 $\mathrm{Q}$ 点). 已知 $R_{3}=2 R_{1}$, 则下列关系正确的是 ( ) [图1] A: $T_{3}=2 \sqrt{2} T_{1}$ B: $T_{2}=\frac{3 \sqrt{6}}{8} T_{1}$ C: $T_{2}=\frac{3 \sqrt{3}}{8} T_{3}$ D: $T_{3}=\frac{3}{4} \sqrt{6} T_{1}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 如图所示, 某次发射远地圆轨道卫星时, 先让卫星进入一个近地的圆轨道I, 在此轨道运行的卫星的轨道半径为 $R_{1}$ 、周期为 $T_{1}$; 然后在 $P$ 点点火加速, 进入陏圆形转移轨道II, 在此轨道运行的卫星的周期为 $T_{2}$; 到达远地点 $\mathrm{Q}$ 时再次点火加速, 进入远地圆轨道III, 在此轨道运行的卫星的轨道半径为 $R_{3}$ 、周期为 $T_{3}$ (轨道II的近地点为 $\mathrm{I}$ 上的 $\mathrm{P}$ 点,远地点为轨道III上的 $\mathrm{Q}$ 点). 已知 $R_{3}=2 R_{1}$, 则下列关系正确的是 ( ) [图1] A: $T_{3}=2 \sqrt{2} T_{1}$ B: $T_{2}=\frac{3 \sqrt{6}}{8} T_{1}$ C: $T_{2}=\frac{3 \sqrt{3}}{8} T_{3}$ D: $T_{3}=\frac{3}{4} \sqrt{6} T_{1}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

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Astronomy

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multi-modal

Astronomy_538

经国际小行星命名委员会命名的“神舟星”和“杨利伟星”的轨道均处在火星和木星轨道之间, 已知“神舟星”平均每天绕太阳运行 174 万公里, “杨利伟星”平均每天绕太阳运行 145 万公里, 假设两行星均绕太阳做匀速圆周运动, 则两星相比较 ( ) A: “神舟星”的轨道半径大 B: “神舟星”的公转周期大 C: “神舟星”的加速度大 D: “神舟星”受到的向心力大

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 经国际小行星命名委员会命名的“神舟星”和“杨利伟星”的轨道均处在火星和木星轨道之间, 已知“神舟星”平均每天绕太阳运行 174 万公里, “杨利伟星”平均每天绕太阳运行 145 万公里, 假设两行星均绕太阳做匀速圆周运动, 则两星相比较 ( ) A: “神舟星”的轨道半径大 B: “神舟星”的公转周期大 C: “神舟星”的加速度大 D: “神舟星”受到的向心力大 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy_895

When Mars has its opposition in 2022, an observer in the UK will see it in Taurus. In roughly which month will this opposition take place? [Mars' opposition corresponds to when it is closest in its orbit to the Earth.] A: March B: June C: September D: December

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: When Mars has its opposition in 2022, an observer in the UK will see it in Taurus. In roughly which month will this opposition take place? [Mars' opposition corresponds to when it is closest in its orbit to the Earth.] A: March B: June C: September D: December You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

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Astronomy

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text-only

Astronomy_568

太阳主要是由 e、 ${ }_{1}^{1} \mathrm{H}$ 和 ${ }_{2}^{4} \mathrm{He}$ 等粒子组成的。维持太阳辐射的是其内部的核聚变反应, 核反应方程是 $2 \mathrm{e}+4{ }_{1}^{1} \mathrm{H} \rightarrow{ }_{2}^{4} \mathrm{He}$, 该核反应产生的核能最后转化为辐射能。根据目前关于 恒星演化的理论, 若由于聚变反应而使太阳中的 ${ }_{1}^{1} \mathrm{H}$ 核数目从现有数减少 $10 \%$, 太阳将离开主序星阶段而转入红巨星的演化阶段。为了简化模型, 假定目前太阳全部由 $\mathrm{e}$ 和 ${ }_{1}{ }^{1} \mathrm{H}$核组成, 并据此回答下列问题。 已知地球半径 $R=6.4 \times 10^{6} \mathrm{~m}$, 地球质量 $m=6.0 \times 10^{24} \mathrm{~kg}$, 日地中心的距离 $r=1.5 \times 10^{11} \mathrm{~m}$, 地球表面处的重力加速度 $g=10 \mathrm{~m} / \mathrm{s}^{2}, 1$ 年约为 $3.2 \times 10^{7} \mathrm{~s}$, 已知质子质量 $m_{\mathrm{p}}=1.6726 \times 10^{-27} \mathrm{~kg},{ }_{2}^{4} \mathrm{He}$ 质量 $m_{\alpha}=6.6458 \times 10^{-27} \mathrm{~kg}$, 电子质量 $m_{\mathrm{e}}=9.1 \times 10^{-31} \mathrm{~kg}$, 光速 $c=3 \times 10^{8} \mathrm{~m} / \mathrm{s}$ 。求题中所述的核聚变反应所释放的核能。

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个数值。 问题: 太阳主要是由 e、 ${ }_{1}^{1} \mathrm{H}$ 和 ${ }_{2}^{4} \mathrm{He}$ 等粒子组成的。维持太阳辐射的是其内部的核聚变反应, 核反应方程是 $2 \mathrm{e}+4{ }_{1}^{1} \mathrm{H} \rightarrow{ }_{2}^{4} \mathrm{He}$, 该核反应产生的核能最后转化为辐射能。根据目前关于 恒星演化的理论, 若由于聚变反应而使太阳中的 ${ }_{1}^{1} \mathrm{H}$ 核数目从现有数减少 $10 \%$, 太阳将离开主序星阶段而转入红巨星的演化阶段。为了简化模型, 假定目前太阳全部由 $\mathrm{e}$ 和 ${ }_{1}{ }^{1} \mathrm{H}$核组成, 并据此回答下列问题。 已知地球半径 $R=6.4 \times 10^{6} \mathrm{~m}$, 地球质量 $m=6.0 \times 10^{24} \mathrm{~kg}$, 日地中心的距离 $r=1.5 \times 10^{11} \mathrm{~m}$, 地球表面处的重力加速度 $g=10 \mathrm{~m} / \mathrm{s}^{2}, 1$ 年约为 $3.2 \times 10^{7} \mathrm{~s}$, 已知质子质量 $m_{\mathrm{p}}=1.6726 \times 10^{-27} \mathrm{~kg},{ }_{2}^{4} \mathrm{He}$ 质量 $m_{\alpha}=6.6458 \times 10^{-27} \mathrm{~kg}$, 电子质量 $m_{\mathrm{e}}=9.1 \times 10^{-31} \mathrm{~kg}$, 光速 $c=3 \times 10^{8} \mathrm{~m} / \mathrm{s}$ 。求题中所述的核聚变反应所释放的核能。 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 请记住，你的答案应以J为单位计算，但在给出最终答案时，请不要包含单位。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。

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Astronomy_510

在银河系中, 双星系统的数量非常多。研究双星, 不但对于了解恒星形成和演化过程的多样性有重要的意义, 而且对于了解银河系的形成与和演化, 也是一个不可缺少的方面。假设在宇宙中远离其他星体的空间中存在由两个质量分别为 $4 m 、 m$ 的天体 $\mathrm{A} 、 \mathrm{~B}$组成的双星系统, 二者中心间的距离为 $L \circ a 、 b$ 两点为两天体所在直线与天体 $B$ 表面的交点, 天体 $\mathrm{B}$ 的半径为 $\frac{L}{5}$ 。已知引力常量为 $G$, 则 $\mathrm{A} 、 \mathrm{~B}$ 两天体运动的周期和 $a 、 b$ 两点处质量为 $m_{0}$ 的物体（视为质点）所受万有引力大小之差为（） A: $2 \pi \sqrt{\frac{L^{3}}{5 G m}}, 0$ B: $2 \pi \sqrt{\frac{L^{3}}{5 G m}}, \frac{325 G m m_{0}}{36 L^{2}}$ C: $\pi \sqrt{\frac{L^{3}}{5 G m}}, 0$ D: $\pi \sqrt{\frac{L^{3}}{5 G m}}, \frac{325 G m m_{0}}{36 L^{2}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 在银河系中, 双星系统的数量非常多。研究双星, 不但对于了解恒星形成和演化过程的多样性有重要的意义, 而且对于了解银河系的形成与和演化, 也是一个不可缺少的方面。假设在宇宙中远离其他星体的空间中存在由两个质量分别为 $4 m 、 m$ 的天体 $\mathrm{A} 、 \mathrm{~B}$组成的双星系统, 二者中心间的距离为 $L \circ a 、 b$ 两点为两天体所在直线与天体 $B$ 表面的交点, 天体 $\mathrm{B}$ 的半径为 $\frac{L}{5}$ 。已知引力常量为 $G$, 则 $\mathrm{A} 、 \mathrm{~B}$ 两天体运动的周期和 $a 、 b$ 两点处质量为 $m_{0}$ 的物体（视为质点）所受万有引力大小之差为（） A: $2 \pi \sqrt{\frac{L^{3}}{5 G m}}, 0$ B: $2 \pi \sqrt{\frac{L^{3}}{5 G m}}, \frac{325 G m m_{0}}{36 L^{2}}$ C: $\pi \sqrt{\frac{L^{3}}{5 G m}}, 0$ D: $\pi \sqrt{\frac{L^{3}}{5 G m}}, \frac{325 G m m_{0}}{36 L^{2}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

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Astronomy_658

2011 年 11 月 3 日凌晨, “神舟八号”飞船与“天宫一号”空间站成功对接。对接后,空间站在离地面三百多公里的轨道上绕地球做匀速圆周运动。现已测出其绕地球球心做匀速圆周运动的周期为 $T$, 已知地球半径为 $R$ 、地球表面重力加速度 $g$ 、万有引力常量为 $G$ ，则根据以上数据能够计算的物理量是 A: 地球的平均密度 B: 空间站所在处的重力加速度大小 C: 空间站绕行的线速度大小 D: 空间站所受的万有引力大小

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 2011 年 11 月 3 日凌晨, “神舟八号”飞船与“天宫一号”空间站成功对接。对接后,空间站在离地面三百多公里的轨道上绕地球做匀速圆周运动。现已测出其绕地球球心做匀速圆周运动的周期为 $T$, 已知地球半径为 $R$ 、地球表面重力加速度 $g$ 、万有引力常量为 $G$ ，则根据以上数据能够计算的物理量是 A: 地球的平均密度 B: 空间站所在处的重力加速度大小 C: 空间站绕行的线速度大小 D: 空间站所受的万有引力大小 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

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Astronomy

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Astronomy_1045

The Sun is our closest star so it is arguably the most studied. Results from detailed observations of how sound waves propagate through the plasma of the Sun allow us to get a sense of the general structure of the Sun, with an upper layer of convecting plasma (leading to the 'bubbling' we see with granulation on the surface) and radiative heat transfer below, including a central core region where the pressure and temperature are large enough for nuclear fusion to occur (see Figure 3). [figure1] Figure 3: Left: The general structure of the Sun, with a core in which all the nuclear reactions take place, and convective cells in an outer layer. Credit: Dmitri Pogosyan / University of Alberta. Right: The fraction of the Sun's mass and the contribution to the Sun's luminosity as a function of solar radius as determined from detailed computer simulations. Essentially all of the nuclear reactions creating the photons for the Sun's luminosity take place in a core with a radius of $0.20 R_{\odot}$ and a mass of $0.35 M_{\odot}$. Credit: Kevin France / University of Colorado. Estimating the conditions in the cores of stars is an important aspect of constructing stellar models. This question explores some of the equations governing stellar structure and estimates the central temperature and pressure of the Sun. The primary nuclear fusion pathway responsible for much of the Sun's luminous output is called the proton-proton chain (p-p chain). All of the most common steps are shown in Figure 4. [figure2] Figure 4: An overview of the all the steps in the most common form of the p-p chain, turning four protons into one helium-4 nucleus plus some light particles and photons. Credit: Wikipedia. The net reaction of the p-p chain is $$ 4_{1}^{1} \mathrm{H} \rightarrow{ }_{2}^{4} \mathrm{He}+2 e^{+}+2 \nu_{e}+2 \gamma . $$ The rate-limiting step is the first one $\left({ }_{1}^{1} \mathrm{H}+{ }_{1}^{1} \mathrm{H}\right)$ as the probability of interaction is so low, since it requires the decay of a proton into a neutron and so involves the weak nuclear force. Nuclear physics allow us to calculate the reaction rate coefficient, $R$, energy produced per reaction, $Q$, and hence the energy generation rate, $q$. The reaction rate is related to the number of particles that have enough energy to undergo quantum tunnelling, and the distribution as a function of energy is known as the Gamow peak, with the top of the curve at energy $E_{0}$. For two nuclei, ${ }_{Z_{i}}^{A_{i}} C_{i}$ and ${ }_{Z_{j}}^{A_{j}} C_{j}$, with mass fractions $X_{i}$ and $X_{j}$, then to first order and ignoring electron screening, $$ R=\frac{4}{3^{2.5} \pi^{2}} \frac{h}{\mu_{r} m_{p}} \frac{4 \pi \varepsilon_{0}}{Z_{i} Z_{j} e^{2}} S\left(E_{0}\right) \tau^{2} e^{-\tau}, \quad \text { where } \quad \mu_{r}=\frac{A_{i} A_{j}}{A_{i}+A_{j}} $$ and $S\left(E_{0}\right)$ measures the probability of interaction at the maximum of the Gamow peak whilst $\tau$ is a characteristic width of the Gamow peak, $$ \tau=\frac{3 E_{0}}{k_{B} T} \quad \text { where } \quad E_{0}=\left(\frac{b k_{B} T}{2}\right)^{2 / 3} \quad \text { given } \quad b=\sqrt{\frac{\mu_{r} m_{p}}{2}} \frac{\pi Z_{i} Z_{j} e^{2}}{h \varepsilon_{0}} $$ Here $h$ is Planck's constant, $\varepsilon_{0}$ is the permittivity of free space and $e$ is the elementary charge (the charge on a proton). Finally, the energy generation rate per unit mass is $$ q=\frac{\rho}{m_{p}^{2}}\left(\frac{1}{1+\delta_{i j}}\right) \frac{X_{i} X_{j}}{A_{i} A_{j}} R Q $$ where $\delta_{i j}$ is the Kronecker delta, so equals 1 when $i=j$ and 0 otherwise. Evaluating the fundamental constants and defining $T_{6} \equiv \frac{T}{10^{6} \mathrm{~K}}$ gives $$ \tau=42.59\left[Z_{i}^{2} Z_{j}^{2} \mu_{r} T_{6}^{-1}\right]^{1 / 3}, $$ whilst for the proton-proton interaction $Q=13.366 \mathrm{MeV}$ (half the overall energy of the p-p chain), $Z_{i}=Z_{j}=A_{i}=A_{j}=\delta_{i j}=1$, and $S\left(E_{0}\right)$ is $4.01 \times 10^{-50} \mathrm{keV} \mathrm{m}^{2}$ (Adelberger et al. 2011), so $R=6.55 \times 10^{-43} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \mathrm{~m}^{3} \mathrm{~s}^{-1}$ and $q=0.251 \rho X^{2} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \mathrm{~W} \mathrm{~kg}^{-1}$.c. Considering the evaluated equations for $\tau, R$, and $q$ we can use this with the measured luminosity of the Sun to get a new estimate for the central temperature. i. Considering the simplified equation for $q$ and assuming that the core has a mass of $0.35 \mathrm{M}_{\odot}$, throughout which T and $\rho$ are constant, and that the Sun's luminosity is equal to the power produced by the $p-p$ chain fusion processes occurring within its core, estimate the central temperature. [You are given that $u=3 p_{c} / 2 p_{c}$.]

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: The Sun is our closest star so it is arguably the most studied. Results from detailed observations of how sound waves propagate through the plasma of the Sun allow us to get a sense of the general structure of the Sun, with an upper layer of convecting plasma (leading to the 'bubbling' we see with granulation on the surface) and radiative heat transfer below, including a central core region where the pressure and temperature are large enough for nuclear fusion to occur (see Figure 3). [figure1] Figure 3: Left: The general structure of the Sun, with a core in which all the nuclear reactions take place, and convective cells in an outer layer. Credit: Dmitri Pogosyan / University of Alberta. Right: The fraction of the Sun's mass and the contribution to the Sun's luminosity as a function of solar radius as determined from detailed computer simulations. Essentially all of the nuclear reactions creating the photons for the Sun's luminosity take place in a core with a radius of $0.20 R_{\odot}$ and a mass of $0.35 M_{\odot}$. Credit: Kevin France / University of Colorado. Estimating the conditions in the cores of stars is an important aspect of constructing stellar models. This question explores some of the equations governing stellar structure and estimates the central temperature and pressure of the Sun. The primary nuclear fusion pathway responsible for much of the Sun's luminous output is called the proton-proton chain (p-p chain). All of the most common steps are shown in Figure 4. [figure2] Figure 4: An overview of the all the steps in the most common form of the p-p chain, turning four protons into one helium-4 nucleus plus some light particles and photons. Credit: Wikipedia. The net reaction of the p-p chain is $$ 4_{1}^{1} \mathrm{H} \rightarrow{ }_{2}^{4} \mathrm{He}+2 e^{+}+2 \nu_{e}+2 \gamma . $$ The rate-limiting step is the first one $\left({ }_{1}^{1} \mathrm{H}+{ }_{1}^{1} \mathrm{H}\right)$ as the probability of interaction is so low, since it requires the decay of a proton into a neutron and so involves the weak nuclear force. Nuclear physics allow us to calculate the reaction rate coefficient, $R$, energy produced per reaction, $Q$, and hence the energy generation rate, $q$. The reaction rate is related to the number of particles that have enough energy to undergo quantum tunnelling, and the distribution as a function of energy is known as the Gamow peak, with the top of the curve at energy $E_{0}$. For two nuclei, ${ }_{Z_{i}}^{A_{i}} C_{i}$ and ${ }_{Z_{j}}^{A_{j}} C_{j}$, with mass fractions $X_{i}$ and $X_{j}$, then to first order and ignoring electron screening, $$ R=\frac{4}{3^{2.5} \pi^{2}} \frac{h}{\mu_{r} m_{p}} \frac{4 \pi \varepsilon_{0}}{Z_{i} Z_{j} e^{2}} S\left(E_{0}\right) \tau^{2} e^{-\tau}, \quad \text { where } \quad \mu_{r}=\frac{A_{i} A_{j}}{A_{i}+A_{j}} $$ and $S\left(E_{0}\right)$ measures the probability of interaction at the maximum of the Gamow peak whilst $\tau$ is a characteristic width of the Gamow peak, $$ \tau=\frac{3 E_{0}}{k_{B} T} \quad \text { where } \quad E_{0}=\left(\frac{b k_{B} T}{2}\right)^{2 / 3} \quad \text { given } \quad b=\sqrt{\frac{\mu_{r} m_{p}}{2}} \frac{\pi Z_{i} Z_{j} e^{2}}{h \varepsilon_{0}} $$ Here $h$ is Planck's constant, $\varepsilon_{0}$ is the permittivity of free space and $e$ is the elementary charge (the charge on a proton). Finally, the energy generation rate per unit mass is $$ q=\frac{\rho}{m_{p}^{2}}\left(\frac{1}{1+\delta_{i j}}\right) \frac{X_{i} X_{j}}{A_{i} A_{j}} R Q $$ where $\delta_{i j}$ is the Kronecker delta, so equals 1 when $i=j$ and 0 otherwise. Evaluating the fundamental constants and defining $T_{6} \equiv \frac{T}{10^{6} \mathrm{~K}}$ gives $$ \tau=42.59\left[Z_{i}^{2} Z_{j}^{2} \mu_{r} T_{6}^{-1}\right]^{1 / 3}, $$ whilst for the proton-proton interaction $Q=13.366 \mathrm{MeV}$ (half the overall energy of the p-p chain), $Z_{i}=Z_{j}=A_{i}=A_{j}=\delta_{i j}=1$, and $S\left(E_{0}\right)$ is $4.01 \times 10^{-50} \mathrm{keV} \mathrm{m}^{2}$ (Adelberger et al. 2011), so $R=6.55 \times 10^{-43} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \mathrm{~m}^{3} \mathrm{~s}^{-1}$ and $q=0.251 \rho X^{2} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \mathrm{~W} \mathrm{~kg}^{-1}$. problem: c. Considering the evaluated equations for $\tau, R$, and $q$ we can use this with the measured luminosity of the Sun to get a new estimate for the central temperature. i. Considering the simplified equation for $q$ and assuming that the core has a mass of $0.35 \mathrm{M}_{\odot}$, throughout which T and $\rho$ are constant, and that the Sun's luminosity is equal to the power produced by the $p-p$ chain fusion processes occurring within its core, estimate the central temperature. [You are given that $u=3 p_{c} / 2 p_{c}$.] All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of \mathrm{~K}, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

multi-modal

Astronomy_952

Mercury is the innermost of the Solar System's planets and so is most influenced by gravitational interactions with the Sun, tying its rotational period to its orbital period in a similar way to the tidal locking between the Moon and the Earth. It orbits the Sun in a 3:2 resonance, meaning that it rotates on its axis three times for every two orbits of the Sun. [figure1] Figure 1: True colour image of Mercury taken by the probe MESSENGER after its closest approach in 2008. Credit: NASA/Johns Hopkins University Applied Physics Laboratory/Carnegie. Mercury has an orbital period of 88 Earth days and a radius of $2440 \mathrm{~km}$, and spins in the same direction as it orbits (both are anti-clockwise as viewed from high above the Sun). Thinking carefully about the geometry of the situation, calculate the length of a solar day as observed on Mercury (i.e. the length of time from one noon to the next). Hint: you should find it is longer than a Mercurian year.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. problem: Mercury is the innermost of the Solar System's planets and so is most influenced by gravitational interactions with the Sun, tying its rotational period to its orbital period in a similar way to the tidal locking between the Moon and the Earth. It orbits the Sun in a 3:2 resonance, meaning that it rotates on its axis three times for every two orbits of the Sun. [figure1] Figure 1: True colour image of Mercury taken by the probe MESSENGER after its closest approach in 2008. Credit: NASA/Johns Hopkins University Applied Physics Laboratory/Carnegie. Mercury has an orbital period of 88 Earth days and a radius of $2440 \mathrm{~km}$, and spins in the same direction as it orbits (both are anti-clockwise as viewed from high above the Sun). Thinking carefully about the geometry of the situation, calculate the length of a solar day as observed on Mercury (i.e. the length of time from one noon to the next). Hint: you should find it is longer than a Mercurian year. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value.

NV

Astronomy

EN

multi-modal

Astronomy_439

我国航天技术走在世界的前列，探月工程“绕、落、回”三步走的最后一步即将完成,即月球投测器实现采样返回. 如图所示为该过程简化后的示意图, 探测器从圆轨道 1 上的 $\mathrm{A}$ 点减速后变轨到椭圆轨道 2 , 之后又在轨道 2 上的 $\mathrm{B}$ 点变轨到近月圆轨道 3 . 已知探测器在轨道 1 上的运行周期为 $T_{1}, O$ 为月球球心, $\mathrm{C}$ 为轨道 3 上的一点, $\mathrm{AC}$ 与 $\mathrm{AO}$之间的最大夹角为 $\theta$. 下列说法正确的是 ( ) [图1] A: 探测器在轨道 2 运行时的机械能大于在轨道 1 运行时的机械能 B: 探测器在轨道 $1 、 2 、 3$ 运行时的周期大小关系为 $\mathrm{T}_{1}<\mathrm{T}_{2}<\mathrm{T}_{3}$ C: 探测器在轨道 2 上运行和在圆轨道 1 上运行, 加速度大小相等的位置有两个 D: 探测器在轨道 3 上运行时的周期为 $\sqrt{\sin ^{3} \theta} \mathrm{T}_{1}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 我国航天技术走在世界的前列，探月工程“绕、落、回”三步走的最后一步即将完成,即月球投测器实现采样返回. 如图所示为该过程简化后的示意图, 探测器从圆轨道 1 上的 $\mathrm{A}$ 点减速后变轨到椭圆轨道 2 , 之后又在轨道 2 上的 $\mathrm{B}$ 点变轨到近月圆轨道 3 . 已知探测器在轨道 1 上的运行周期为 $T_{1}, O$ 为月球球心, $\mathrm{C}$ 为轨道 3 上的一点, $\mathrm{AC}$ 与 $\mathrm{AO}$之间的最大夹角为 $\theta$. 下列说法正确的是 ( ) [图1] A: 探测器在轨道 2 运行时的机械能大于在轨道 1 运行时的机械能 B: 探测器在轨道 $1 、 2 、 3$ 运行时的周期大小关系为 $\mathrm{T}_{1}<\mathrm{T}_{2}<\mathrm{T}_{3}$ C: 探测器在轨道 2 上运行和在圆轨道 1 上运行, 加速度大小相等的位置有两个 D: 探测器在轨道 3 上运行时的周期为 $\sqrt{\sin ^{3} \theta} \mathrm{T}_{1}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

multi-modal

Astronomy_1081

The Sun is our closest star so it is arguably the most studied. Results from detailed observations of how sound waves propagate through the plasma of the Sun allow us to get a sense of the general structure of the Sun, with an upper layer of convecting plasma (leading to the 'bubbling' we see with granulation on the surface) and radiative heat transfer below, including a central core region where the pressure and temperature are large enough for nuclear fusion to occur (see Figure 3). [figure1] Figure 3: Left: The general structure of the Sun, with a core in which all the nuclear reactions take place, and convective cells in an outer layer. Credit: Dmitri Pogosyan / University of Alberta. Right: The fraction of the Sun's mass and the contribution to the Sun's luminosity as a function of solar radius as determined from detailed computer simulations. Essentially all of the nuclear reactions creating the photons for the Sun's luminosity take place in a core with a radius of $0.20 R_{\odot}$ and a mass of $0.35 M_{\odot}$. Credit: Kevin France / University of Colorado. Estimating the conditions in the cores of stars is an important aspect of constructing stellar models. This question explores some of the equations governing stellar structure and estimates the central temperature and pressure of the Sun. The primary nuclear fusion pathway responsible for much of the Sun's luminous output is called the proton-proton chain (p-p chain). All of the most common steps are shown in Figure 4. [figure2] Figure 4: An overview of the all the steps in the most common form of the p-p chain, turning four protons into one helium-4 nucleus plus some light particles and photons. Credit: Wikipedia. The net reaction of the p-p chain is $$ 4_{1}^{1} \mathrm{H} \rightarrow{ }_{2}^{4} \mathrm{He}+2 e^{+}+2 \nu_{e}+2 \gamma . $$ The rate-limiting step is the first one $\left({ }_{1}^{1} \mathrm{H}+{ }_{1}^{1} \mathrm{H}\right)$ as the probability of interaction is so low, since it requires the decay of a proton into a neutron and so involves the weak nuclear force. Nuclear physics allow us to calculate the reaction rate coefficient, $R$, energy produced per reaction, $Q$, and hence the energy generation rate, $q$. The reaction rate is related to the number of particles that have enough energy to undergo quantum tunnelling, and the distribution as a function of energy is known as the Gamow peak, with the top of the curve at energy $E_{0}$. For two nuclei, ${ }_{Z_{i}}^{A_{i}} C_{i}$ and ${ }_{Z_{j}}^{A_{j}} C_{j}$, with mass fractions $X_{i}$ and $X_{j}$, then to first order and ignoring electron screening, $$ R=\frac{4}{3^{2.5} \pi^{2}} \frac{h}{\mu_{r} m_{p}} \frac{4 \pi \varepsilon_{0}}{Z_{i} Z_{j} e^{2}} S\left(E_{0}\right) \tau^{2} e^{-\tau}, \quad \text { where } \quad \mu_{r}=\frac{A_{i} A_{j}}{A_{i}+A_{j}} $$ and $S\left(E_{0}\right)$ measures the probability of interaction at the maximum of the Gamow peak whilst $\tau$ is a characteristic width of the Gamow peak, $$ \tau=\frac{3 E_{0}}{k_{B} T} \quad \text { where } \quad E_{0}=\left(\frac{b k_{B} T}{2}\right)^{2 / 3} \quad \text { given } \quad b=\sqrt{\frac{\mu_{r} m_{p}}{2}} \frac{\pi Z_{i} Z_{j} e^{2}}{h \varepsilon_{0}} $$ Here $h$ is Planck's constant, $\varepsilon_{0}$ is the permittivity of free space and $e$ is the elementary charge (the charge on a proton). Finally, the energy generation rate per unit mass is $$ q=\frac{\rho}{m_{p}^{2}}\left(\frac{1}{1+\delta_{i j}}\right) \frac{X_{i} X_{j}}{A_{i} A_{j}} R Q $$ where $\delta_{i j}$ is the Kronecker delta, so equals 1 when $i=j$ and 0 otherwise. Evaluating the fundamental constants and defining $T_{6} \equiv \frac{T}{10^{6} \mathrm{~K}}$ gives $$ \tau=42.59\left[Z_{i}^{2} Z_{j}^{2} \mu_{r} T_{6}^{-1}\right]^{1 / 3}, $$ whilst for the proton-proton interaction $Q=13.366 \mathrm{MeV}$ (half the overall energy of the p-p chain), $Z_{i}=Z_{j}=A_{i}=A_{j}=\delta_{i j}=1$, and $S\left(E_{0}\right)$ is $4.01 \times 10^{-50} \mathrm{keV} \mathrm{m}^{2}$ (Adelberger et al. 2011), so $R=6.55 \times 10^{-43} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \mathrm{~m}^{3} \mathrm{~s}^{-1}$ and $q=0.251 \rho X^{2} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \mathrm{~W} \mathrm{~kg}^{-1}$.a. Let $r$ denote distance from the centre of a star. We define the variables $\rho(r), p(r)$ and $T(r)$ to be the density, pressure and temperature at radius $r$ respectively, and $m(r)$ to be the mass enclosed within radius $r$. We will now try and derive an estimate for the pressure at the centre of the Sun. iii. Assuming that the pressure at the surface, $\mathrm{p}_{\mathrm{s}}$, is negligible compared to the pressure at the centre of the Sun, $p_{c}$, the edge of the core is at $r=0.20 R_{\odot}$ and encloses a mass of $m=0.35$ $\mathrm{M}_{\odot}$, and that $\mathrm{dp} / \mathrm{dm}$ is constant throughout the star and equal to the value at the edge of the core, calculate a value for $\mathrm{p}_{\mathrm{c}}$.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: The Sun is our closest star so it is arguably the most studied. Results from detailed observations of how sound waves propagate through the plasma of the Sun allow us to get a sense of the general structure of the Sun, with an upper layer of convecting plasma (leading to the 'bubbling' we see with granulation on the surface) and radiative heat transfer below, including a central core region where the pressure and temperature are large enough for nuclear fusion to occur (see Figure 3). [figure1] Figure 3: Left: The general structure of the Sun, with a core in which all the nuclear reactions take place, and convective cells in an outer layer. Credit: Dmitri Pogosyan / University of Alberta. Right: The fraction of the Sun's mass and the contribution to the Sun's luminosity as a function of solar radius as determined from detailed computer simulations. Essentially all of the nuclear reactions creating the photons for the Sun's luminosity take place in a core with a radius of $0.20 R_{\odot}$ and a mass of $0.35 M_{\odot}$. Credit: Kevin France / University of Colorado. Estimating the conditions in the cores of stars is an important aspect of constructing stellar models. This question explores some of the equations governing stellar structure and estimates the central temperature and pressure of the Sun. The primary nuclear fusion pathway responsible for much of the Sun's luminous output is called the proton-proton chain (p-p chain). All of the most common steps are shown in Figure 4. [figure2] Figure 4: An overview of the all the steps in the most common form of the p-p chain, turning four protons into one helium-4 nucleus plus some light particles and photons. Credit: Wikipedia. The net reaction of the p-p chain is $$ 4_{1}^{1} \mathrm{H} \rightarrow{ }_{2}^{4} \mathrm{He}+2 e^{+}+2 \nu_{e}+2 \gamma . $$ The rate-limiting step is the first one $\left({ }_{1}^{1} \mathrm{H}+{ }_{1}^{1} \mathrm{H}\right)$ as the probability of interaction is so low, since it requires the decay of a proton into a neutron and so involves the weak nuclear force. Nuclear physics allow us to calculate the reaction rate coefficient, $R$, energy produced per reaction, $Q$, and hence the energy generation rate, $q$. The reaction rate is related to the number of particles that have enough energy to undergo quantum tunnelling, and the distribution as a function of energy is known as the Gamow peak, with the top of the curve at energy $E_{0}$. For two nuclei, ${ }_{Z_{i}}^{A_{i}} C_{i}$ and ${ }_{Z_{j}}^{A_{j}} C_{j}$, with mass fractions $X_{i}$ and $X_{j}$, then to first order and ignoring electron screening, $$ R=\frac{4}{3^{2.5} \pi^{2}} \frac{h}{\mu_{r} m_{p}} \frac{4 \pi \varepsilon_{0}}{Z_{i} Z_{j} e^{2}} S\left(E_{0}\right) \tau^{2} e^{-\tau}, \quad \text { where } \quad \mu_{r}=\frac{A_{i} A_{j}}{A_{i}+A_{j}} $$ and $S\left(E_{0}\right)$ measures the probability of interaction at the maximum of the Gamow peak whilst $\tau$ is a characteristic width of the Gamow peak, $$ \tau=\frac{3 E_{0}}{k_{B} T} \quad \text { where } \quad E_{0}=\left(\frac{b k_{B} T}{2}\right)^{2 / 3} \quad \text { given } \quad b=\sqrt{\frac{\mu_{r} m_{p}}{2}} \frac{\pi Z_{i} Z_{j} e^{2}}{h \varepsilon_{0}} $$ Here $h$ is Planck's constant, $\varepsilon_{0}$ is the permittivity of free space and $e$ is the elementary charge (the charge on a proton). Finally, the energy generation rate per unit mass is $$ q=\frac{\rho}{m_{p}^{2}}\left(\frac{1}{1+\delta_{i j}}\right) \frac{X_{i} X_{j}}{A_{i} A_{j}} R Q $$ where $\delta_{i j}$ is the Kronecker delta, so equals 1 when $i=j$ and 0 otherwise. Evaluating the fundamental constants and defining $T_{6} \equiv \frac{T}{10^{6} \mathrm{~K}}$ gives $$ \tau=42.59\left[Z_{i}^{2} Z_{j}^{2} \mu_{r} T_{6}^{-1}\right]^{1 / 3}, $$ whilst for the proton-proton interaction $Q=13.366 \mathrm{MeV}$ (half the overall energy of the p-p chain), $Z_{i}=Z_{j}=A_{i}=A_{j}=\delta_{i j}=1$, and $S\left(E_{0}\right)$ is $4.01 \times 10^{-50} \mathrm{keV} \mathrm{m}^{2}$ (Adelberger et al. 2011), so $R=6.55 \times 10^{-43} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \mathrm{~m}^{3} \mathrm{~s}^{-1}$ and $q=0.251 \rho X^{2} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \mathrm{~W} \mathrm{~kg}^{-1}$. problem: a. Let $r$ denote distance from the centre of a star. We define the variables $\rho(r), p(r)$ and $T(r)$ to be the density, pressure and temperature at radius $r$ respectively, and $m(r)$ to be the mass enclosed within radius $r$. We will now try and derive an estimate for the pressure at the centre of the Sun. iii. Assuming that the pressure at the surface, $\mathrm{p}_{\mathrm{s}}$, is negligible compared to the pressure at the centre of the Sun, $p_{c}$, the edge of the core is at $r=0.20 R_{\odot}$ and encloses a mass of $m=0.35$ $\mathrm{M}_{\odot}$, and that $\mathrm{dp} / \mathrm{dm}$ is constant throughout the star and equal to the value at the edge of the core, calculate a value for $\mathrm{p}_{\mathrm{c}}$. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of \mathrm{~Pa}, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

multi-modal

Astronomy_1155

In November 2020 the Aricebo Telescope at the National Astronomy and Ionosphere Centre (NAIC) in Puerto Rico was decommissioned due to safety concerns after extensive storm damage. First opened in November 1963, this brought an end to an illustrious contribution to radio astronomy where, with a dish diameter of $304.8 \mathrm{~m}$ (1000 ft), it was the largest radio telescope in the world until 2016. Its important discoveries range from detection of the first extrasolar planets around a pulsar to fast radio bursts, as well as a pivotal role in the search for extraterrestrial intelligence (SETI), however in this question we will explore its earliest major revelation: that Mercury was not tidally locked. [figure1] Figure 1: Left: The Aricebo telescope before it was damaged. Credit: NAIC. Right: When transmitting a pulse from a radio telescope, diffraction prevents the beam from staying perfectly parallel and so the width of the beam increases by $2 \theta$. Credit: OpenStax, College Physics. Mercury had already been studied with optical and infrared telescopes, however the advantage of a radio telescope was that you could send pulses and receive their reflections. This radar-ranging technique had already been used with Venus to measure the distance to it and hence provide the data necessary for a definitive measurement of an astronomical unit in metres. The radar echo from Mercury is much harder to detect due to the extra distance travelled and its smaller cross-sectional area (its radius is $2440 \mathrm{~km}$ ). In April 1965, Pettengill and Dyce sent a series of $500 \mu \mathrm{s}$ pulses at $430 \mathrm{MHz}$ with a transmitted power of $2.0 \mathrm{MW}$ towards Mercury whilst it was at its closest point in its orbit to Earth. In ideal circumstances the beam would stay parallel, however diffraction widens the beam as shown on the right in Fig 1. The signal-to-noise ratio of this echo was high enough that Doppler broadening of the received signal was reliably detected, allowing a determination of the rotation rate of Mercury. In August 1965 the same scientists sent $100 \mu$ s pulses and sampled the echo on short timescales as it returned. The strongest echo (received first) came from the point of the planet closest to the Earth (called the sub-radar point), with later echos coming from other parts of the surface in an annulus of increasing radius (see Fig 2). Photons from the approaching side would be blueshifted to a higher frequency, whilst those from the receding side would be redshifted to a lower frequency. Hence, by measuring the Doppler shift and the time delay, you can map the rotational velocity as a function of apparent longitude and so can calculate the apparent rotation rate (as well as the direction of rotation and co-ordinates of the pole). [figure2] Figure 2: Left: Snapshots of the reflections of a single $100 \mu$ sulse. The strength of the echo weakens as you move to later delays (as represented by the scale factor in the top right of each snapshot) and hence you need to use an annulus rather than detections from the horizon. The small arrows indicate the Doppler shifted frequency associated with intersection of the annulus with the apparent equator for each delay. The horizontal axis is in cycles per second (and so $1 \mathrm{c} / \mathrm{s}=1 \mathrm{~Hz}$ ). Credit: Dyce, Pettengill and Shapiro (1967). Top right: The key principles of the delay-Doppler technique, looking at a cross-section of the planet. At the very centre is the sub-radar point (the point on the planet's surface closest to the Earth). As you move away from the sub-radar point the light has to travel further before it can be reflected, and hence the echo from those regions arrives later. The brightest point of any given annulus is where it intersects the apparent equator (due to the largest reflecting area), and so in each of the snapshots that is why the extreme Doppler shifts are boosted relative to the middle. Credit: Shapiro (1967). Bottom right: The same as the snapshots, but this time summed over the first $500 \mu$ of reflections. Here the difference between the extreme left and right frequencies reliably detected is $\sim 5 \mathrm{~Hz}$, but when corrected for relative motion of the Earth and Mercury it becomes the value given in part c. Credit: Pettengill, Dyce and Campbell (1967). The Doppler shift with light is given as $$ \frac{\Delta f}{f}=\frac{v}{c} $$ where $\Delta f$ is the shift in frequency $f, v$ is the line-of-sight velocity of the emitting object and $c$ is the speed of light. Ever since the first maps of Mercury's surface by Schiaparelli in the late 1880s, many in the scientific community believed that Mercury would be tidally locked and so always present the same hemisphere to the Sun. The reason they expected the rotational period to be the same as its orbital period (i.e. a $1: 1$ ratio), rather like the Moon, is because it is so close to the Sun and the tidal torques causing this synchronicity are proportional to $r^{-6}$ where $r$ is the distance from the massive body. Given it is the closest planet to the Sun, it receives by far the largest torques, so the discovery it was in a different ratio was a complete surprise to many of the scientists at the time. [figure3] Figure 3: The orientation of Mercury's axis of minimum moment of inertia (the axis the tidal torque acts upon) displayed at six points in its orbit (equally spaced in time) if the ratio had been $1: 1$. Credit: Colombo and Shapiro (1966).a. Calculate the power of each echo received by the Aricebo telescope and hence determine the total number of photons in each echo, given the echo was detected $579.3 \mathrm{~s}$ after being transmitted and Mercury's surface only reflects $6.5 \%$ of the incident radio photons. Assume $\theta=0.16^{\circ}$ and the reflected photons from Mercury are scattered randomly within only the hemisphere facing Earth.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is an expression. Here is some context information for this question, which might assist you in solving it: In November 2020 the Aricebo Telescope at the National Astronomy and Ionosphere Centre (NAIC) in Puerto Rico was decommissioned due to safety concerns after extensive storm damage. First opened in November 1963, this brought an end to an illustrious contribution to radio astronomy where, with a dish diameter of $304.8 \mathrm{~m}$ (1000 ft), it was the largest radio telescope in the world until 2016. Its important discoveries range from detection of the first extrasolar planets around a pulsar to fast radio bursts, as well as a pivotal role in the search for extraterrestrial intelligence (SETI), however in this question we will explore its earliest major revelation: that Mercury was not tidally locked. [figure1] Figure 1: Left: The Aricebo telescope before it was damaged. Credit: NAIC. Right: When transmitting a pulse from a radio telescope, diffraction prevents the beam from staying perfectly parallel and so the width of the beam increases by $2 \theta$. Credit: OpenStax, College Physics. Mercury had already been studied with optical and infrared telescopes, however the advantage of a radio telescope was that you could send pulses and receive their reflections. This radar-ranging technique had already been used with Venus to measure the distance to it and hence provide the data necessary for a definitive measurement of an astronomical unit in metres. The radar echo from Mercury is much harder to detect due to the extra distance travelled and its smaller cross-sectional area (its radius is $2440 \mathrm{~km}$ ). In April 1965, Pettengill and Dyce sent a series of $500 \mu \mathrm{s}$ pulses at $430 \mathrm{MHz}$ with a transmitted power of $2.0 \mathrm{MW}$ towards Mercury whilst it was at its closest point in its orbit to Earth. In ideal circumstances the beam would stay parallel, however diffraction widens the beam as shown on the right in Fig 1. The signal-to-noise ratio of this echo was high enough that Doppler broadening of the received signal was reliably detected, allowing a determination of the rotation rate of Mercury. In August 1965 the same scientists sent $100 \mu$ s pulses and sampled the echo on short timescales as it returned. The strongest echo (received first) came from the point of the planet closest to the Earth (called the sub-radar point), with later echos coming from other parts of the surface in an annulus of increasing radius (see Fig 2). Photons from the approaching side would be blueshifted to a higher frequency, whilst those from the receding side would be redshifted to a lower frequency. Hence, by measuring the Doppler shift and the time delay, you can map the rotational velocity as a function of apparent longitude and so can calculate the apparent rotation rate (as well as the direction of rotation and co-ordinates of the pole). [figure2] Figure 2: Left: Snapshots of the reflections of a single $100 \mu$ sulse. The strength of the echo weakens as you move to later delays (as represented by the scale factor in the top right of each snapshot) and hence you need to use an annulus rather than detections from the horizon. The small arrows indicate the Doppler shifted frequency associated with intersection of the annulus with the apparent equator for each delay. The horizontal axis is in cycles per second (and so $1 \mathrm{c} / \mathrm{s}=1 \mathrm{~Hz}$ ). Credit: Dyce, Pettengill and Shapiro (1967). Top right: The key principles of the delay-Doppler technique, looking at a cross-section of the planet. At the very centre is the sub-radar point (the point on the planet's surface closest to the Earth). As you move away from the sub-radar point the light has to travel further before it can be reflected, and hence the echo from those regions arrives later. The brightest point of any given annulus is where it intersects the apparent equator (due to the largest reflecting area), and so in each of the snapshots that is why the extreme Doppler shifts are boosted relative to the middle. Credit: Shapiro (1967). Bottom right: The same as the snapshots, but this time summed over the first $500 \mu$ of reflections. Here the difference between the extreme left and right frequencies reliably detected is $\sim 5 \mathrm{~Hz}$, but when corrected for relative motion of the Earth and Mercury it becomes the value given in part c. Credit: Pettengill, Dyce and Campbell (1967). The Doppler shift with light is given as $$ \frac{\Delta f}{f}=\frac{v}{c} $$ where $\Delta f$ is the shift in frequency $f, v$ is the line-of-sight velocity of the emitting object and $c$ is the speed of light. Ever since the first maps of Mercury's surface by Schiaparelli in the late 1880s, many in the scientific community believed that Mercury would be tidally locked and so always present the same hemisphere to the Sun. The reason they expected the rotational period to be the same as its orbital period (i.e. a $1: 1$ ratio), rather like the Moon, is because it is so close to the Sun and the tidal torques causing this synchronicity are proportional to $r^{-6}$ where $r$ is the distance from the massive body. Given it is the closest planet to the Sun, it receives by far the largest torques, so the discovery it was in a different ratio was a complete surprise to many of the scientists at the time. [figure3] Figure 3: The orientation of Mercury's axis of minimum moment of inertia (the axis the tidal torque acts upon) displayed at six points in its orbit (equally spaced in time) if the ratio had been $1: 1$. Credit: Colombo and Shapiro (1966). problem: a. Calculate the power of each echo received by the Aricebo telescope and hence determine the total number of photons in each echo, given the echo was detected $579.3 \mathrm{~s}$ after being transmitted and Mercury's surface only reflects $6.5 \%$ of the incident radio photons. Assume $\theta=0.16^{\circ}$ and the reflected photons from Mercury are scattered randomly within only the hemisphere facing Earth. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is an expression without equals signs, e.g. ANSWER=\frac{1}{2} g t^2

EX

Astronomy

EN

multi-modal

Astronomy_1142

On $21^{\text {st }}$ December 2020, Jupiter and Saturn formed a true spectacle in the southwestern sky just after sunset in the UK, separated by only $0.102^{\circ}$. This is close enough that when viewed through a telescope both planets and their moons could be seen in the same field of view (see Fig 4). When two planets occupy the same piece of sky it is known as a conjunction, and when it is Jupiter and Saturn it is known as a great conjunction (so named because they are the rarest of the naked-eye planet conjunctions). The reason it happens is because Jupiter's orbital velocity is higher than Saturn's and so as time goes on Jupiter catches up with and overtakes Saturn (at least as viewed from Earth), with the moment of overtaking corresponding to the conjunction. The process of the two getting closer and closer together has been seen in sky throughout the year (see Fig 5). [figure1] Figure 4: Left: The view of the planets the day before their closest approach, as captured by the 16 " telescope at the Institute of Astronomy, Cambridge. Credit: Robin Catchpole. Right: The view from Arizona on the day of the closest approach as viewed with the Lowell Discovery Telescope. Credit: Levine / Elbert / Bosh / Lowell Observatory. [figure2] Figure 5: Left: Demonstrating how Jupiter and Saturn have been getting closer and closer together over the last few months. Separations are given in degrees and arcminutes $\left(1 / 60^{\text {th }}\right.$ of a degree). Credit: Pete Lawrence. Right: The positions of the Earth, Jupiter and Saturn that were responsible for the 2020 great conjunction. The precise timing of the apparent alignment is clearly sensitive to where Earth is in its orbit. Credit: timeanddate.com. The time between conjunctions is known as the synodic period. Although this period will change slightly from conjunction to conjunction due to different lines of perspective as viewed from Earth (see Fig 5), we can work out the average time between great conjunctions by considering both planets travelling on circular coplanar orbits and ignoring the position of the Earth. For circular coplanar orbits the centre of Jupiter's disc would pass in front of the centre of Saturn's disc every conjunction, and hence have an angular separation of $\theta=0^{\circ}$ (it is measured from the centre of each disc). In practice, the planets follow elliptical orbits that are in planes inclined at different angles to each other. Fig 6 shows how this affects the real values for over 8000 years' worth of data, which along with different synodic periods between conjunctions makes it a difficult problem to solve precisely without a computer. However, after one synodic period Saturn has moved about $2 / 3$ of the way around its orbit, and so roughly every 3 synodic periods it is in a similar part of the sky. Consequently every third great conjunction follows a reasonably regular pattern which can be fit with a sinusoidal function. [figure3] Figure 6: Top: All great conjunctions from 1800 to 2300, calculated for the real celestial mechanics of the Solar System. We can see that each great conjunction belongs to one of three different series or tracks, with Track A indicated with orange circles, Track B with green squares, and Track C with blue triangles. Bottom: The same idea but extended over a much larger date range, up to $10000 \mathrm{AD}$. It is clear the distinct series form broadly sinusoidal patterns which can be used with the average synodic period to give rough predictions for the separations of great conjunctions. The opacity of points is related to each conjunction's angular separation from the Sun (where low opacity means close to the Sun, so it is harder for any observers to see). Credit: Nick Koukoufilippas, but inspired by the work of Steffen Thorsen and Graham Jones / Sky \& Telescope.a. In ideal observing conditions the two planets are far enough apart that they should be (just about) distinguishable to the naked eye, however to some observers in imperfect conditions they would appear as a single bright dot, brighter than either planet on its own. ii. Although they appeared close in angle, there was a very considerable distance between the two planets. At conjunction, Jupiter was 5.926 au from Earth whilst Saturn was 10.827 au (see Fig 5). If they were actually next to each other in space such that they could be treated as a single object, how far from the Earth (in au) would they need to be to have the same apparent magnitude as calculated in the previous part? For simplicity, assume that both planets can be modelled as (very low luminosity) stars so that the change in brightness is only due to changing the distance from the Earth (i.e. ignore the complications from the changing distance from the Sun affecting the number of reflected photons and the changing geometry affecting the illuminated fraction of the planet's surface).

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: On $21^{\text {st }}$ December 2020, Jupiter and Saturn formed a true spectacle in the southwestern sky just after sunset in the UK, separated by only $0.102^{\circ}$. This is close enough that when viewed through a telescope both planets and their moons could be seen in the same field of view (see Fig 4). When two planets occupy the same piece of sky it is known as a conjunction, and when it is Jupiter and Saturn it is known as a great conjunction (so named because they are the rarest of the naked-eye planet conjunctions). The reason it happens is because Jupiter's orbital velocity is higher than Saturn's and so as time goes on Jupiter catches up with and overtakes Saturn (at least as viewed from Earth), with the moment of overtaking corresponding to the conjunction. The process of the two getting closer and closer together has been seen in sky throughout the year (see Fig 5). [figure1] Figure 4: Left: The view of the planets the day before their closest approach, as captured by the 16 " telescope at the Institute of Astronomy, Cambridge. Credit: Robin Catchpole. Right: The view from Arizona on the day of the closest approach as viewed with the Lowell Discovery Telescope. Credit: Levine / Elbert / Bosh / Lowell Observatory. [figure2] Figure 5: Left: Demonstrating how Jupiter and Saturn have been getting closer and closer together over the last few months. Separations are given in degrees and arcminutes $\left(1 / 60^{\text {th }}\right.$ of a degree). Credit: Pete Lawrence. Right: The positions of the Earth, Jupiter and Saturn that were responsible for the 2020 great conjunction. The precise timing of the apparent alignment is clearly sensitive to where Earth is in its orbit. Credit: timeanddate.com. The time between conjunctions is known as the synodic period. Although this period will change slightly from conjunction to conjunction due to different lines of perspective as viewed from Earth (see Fig 5), we can work out the average time between great conjunctions by considering both planets travelling on circular coplanar orbits and ignoring the position of the Earth. For circular coplanar orbits the centre of Jupiter's disc would pass in front of the centre of Saturn's disc every conjunction, and hence have an angular separation of $\theta=0^{\circ}$ (it is measured from the centre of each disc). In practice, the planets follow elliptical orbits that are in planes inclined at different angles to each other. Fig 6 shows how this affects the real values for over 8000 years' worth of data, which along with different synodic periods between conjunctions makes it a difficult problem to solve precisely without a computer. However, after one synodic period Saturn has moved about $2 / 3$ of the way around its orbit, and so roughly every 3 synodic periods it is in a similar part of the sky. Consequently every third great conjunction follows a reasonably regular pattern which can be fit with a sinusoidal function. [figure3] Figure 6: Top: All great conjunctions from 1800 to 2300, calculated for the real celestial mechanics of the Solar System. We can see that each great conjunction belongs to one of three different series or tracks, with Track A indicated with orange circles, Track B with green squares, and Track C with blue triangles. Bottom: The same idea but extended over a much larger date range, up to $10000 \mathrm{AD}$. It is clear the distinct series form broadly sinusoidal patterns which can be used with the average synodic period to give rough predictions for the separations of great conjunctions. The opacity of points is related to each conjunction's angular separation from the Sun (where low opacity means close to the Sun, so it is harder for any observers to see). Credit: Nick Koukoufilippas, but inspired by the work of Steffen Thorsen and Graham Jones / Sky \& Telescope. problem: a. In ideal observing conditions the two planets are far enough apart that they should be (just about) distinguishable to the naked eye, however to some observers in imperfect conditions they would appear as a single bright dot, brighter than either planet on its own. ii. Although they appeared close in angle, there was a very considerable distance between the two planets. At conjunction, Jupiter was 5.926 au from Earth whilst Saturn was 10.827 au (see Fig 5). If they were actually next to each other in space such that they could be treated as a single object, how far from the Earth (in au) would they need to be to have the same apparent magnitude as calculated in the previous part? For simplicity, assume that both planets can be modelled as (very low luminosity) stars so that the change in brightness is only due to changing the distance from the Earth (i.e. ignore the complications from the changing distance from the Sun affecting the number of reflected photons and the changing geometry affecting the illuminated fraction of the planet's surface). All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of au, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

multi-modal

Astronomy_48

如图所示, 横截面积为 $A$ 、质量为 $m$ 的柱状飞行器沿半径为 $R$ 的圆形轨道在高空绕地球做无动力运行。将地球看作质量为 $M$ 的均匀球体。万有引力常量为 $G$ 。 求飞行器在轨道半径为 $R$ 的高空绕地球做圆周运动的线速度; [图1]

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 如图所示, 横截面积为 $A$ 、质量为 $m$ 的柱状飞行器沿半径为 $R$ 的圆形轨道在高空绕地球做无动力运行。将地球看作质量为 $M$ 的均匀球体。万有引力常量为 $G$ 。 求飞行器在轨道半径为 $R$ 的高空绕地球做圆周运动的线速度; [图1] 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

EX

Astronomy

ZH

multi-modal

Astronomy_1179

Why is the Moon heavily cratered, but not the Earth? A: The Moon has stronger gravity, so it attracts more space debris B: The Moon formed earlier than the Earth, so it had more time to be bombarded by asteroids C: The craters on Earth were eroded by the oceans and atmosphere over a long period of time D: The Moon orbits around the Earth in addition to orbiting around the Sun, so it collects more space debris

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: Why is the Moon heavily cratered, but not the Earth? A: The Moon has stronger gravity, so it attracts more space debris B: The Moon formed earlier than the Earth, so it had more time to be bombarded by asteroids C: The craters on Earth were eroded by the oceans and atmosphere over a long period of time D: The Moon orbits around the Earth in addition to orbiting around the Sun, so it collects more space debris You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

SC

Astronomy

EN

text-only

Astronomy_182

质量为 $100 \mathrm{~kg}$ 的“勇气”号火星车于 2004 年成功登陆在火星表面。若“勇气”号在离火星表面 $12 \mathrm{~m}$ 时与降落伞自动脱离, 被气囊包裹的“勇气”号下落到地面后又弹跳到 $18 \mathrm{~m}$ 高处, 这样上下碰撞了若干次后, 才静止在火星表面上。已知火星的半径为地球半径的 0.5 倍, 质量为地球质量的 0.1 倍。若“勇气”号第一次碰撞火星地面时, 气囊和地面的接触时间为 $0.7 \mathrm{~s}$, 其损失的机械能为它与降落伞自动脱离处 (即离火星地面 $12 \mathrm{~m}$ 时) 动能的 70\%, (地球表面的重力加速度 $g=10 \mathrm{~m} / \mathrm{s}^{2}$, 不考虑火星表面空气阻力) 求: “勇气”号在它与降落伞自动脱离处（即离火星地面 $12 \mathrm{~m}$ 时）的速度;

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个数值。 问题: 质量为 $100 \mathrm{~kg}$ 的“勇气”号火星车于 2004 年成功登陆在火星表面。若“勇气”号在离火星表面 $12 \mathrm{~m}$ 时与降落伞自动脱离, 被气囊包裹的“勇气”号下落到地面后又弹跳到 $18 \mathrm{~m}$ 高处, 这样上下碰撞了若干次后, 才静止在火星表面上。已知火星的半径为地球半径的 0.5 倍, 质量为地球质量的 0.1 倍。若“勇气”号第一次碰撞火星地面时, 气囊和地面的接触时间为 $0.7 \mathrm{~s}$, 其损失的机械能为它与降落伞自动脱离处 (即离火星地面 $12 \mathrm{~m}$ 时) 动能的 70\%, (地球表面的重力加速度 $g=10 \mathrm{~m} / \mathrm{s}^{2}$, 不考虑火星表面空气阻力) 求: “勇气”号在它与降落伞自动脱离处（即离火星地面 $12 \mathrm{~m}$ 时）的速度; 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 请记住，你的答案应以m/s为单位计算，但在给出最终答案时，请不要包含单位。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。

NV

Astronomy

ZH

text-only

Astronomy_425

'重力探矿是常用的探测黄金矿藏的方法之一, 是万有引力定律理论的实际应用, 其原理可简述如下：如图, $P 、 Q$ 为某地区水平地面上的两点, 在 $P$ 点正下方一球形区域内充满了富含黄金的矿石，假定球形区域周围普通岩石均匀分布且密度为 $\rho$, 而球形区域内黄金矿石也均匀分布但其密度是普通岩石密度的 $(n+1)$ 倍, 如果没有这一球形区域黄金矿石的存在, 则该地区重力加速度（正常值）沿坚直方向, 当该区域有黄金矿石时, 该地区重力加速度的大小和方向会与正常情况有微小偏离, 重力加速度在原坚直方向（即 $P O$ 方向）上的投影相对于正常值的偏离叫做“重力加速度反常”, 为了探寻黄金矿石区域的位置和储量, 常利用 $P$ 点附近重力加速度反常现象, 已知引力常量为 $G$ 。 设球形区域体积为 $V$, 球心深度为 $d$ ( $d$ 远小于地球半径), $\overline{P Q}=x$, 求: 球形区域内黄金矿石在 $Q$ 点产生的加速度大小; [图1]

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: '重力探矿是常用的探测黄金矿藏的方法之一, 是万有引力定律理论的实际应用, 其原理可简述如下：如图, $P 、 Q$ 为某地区水平地面上的两点, 在 $P$ 点正下方一球形区域内充满了富含黄金的矿石，假定球形区域周围普通岩石均匀分布且密度为 $\rho$, 而球形区域内黄金矿石也均匀分布但其密度是普通岩石密度的 $(n+1)$ 倍, 如果没有这一球形区域黄金矿石的存在, 则该地区重力加速度（正常值）沿坚直方向, 当该区域有黄金矿石时, 该地区重力加速度的大小和方向会与正常情况有微小偏离, 重力加速度在原坚直方向（即 $P O$ 方向）上的投影相对于正常值的偏离叫做“重力加速度反常”, 为了探寻黄金矿石区域的位置和储量, 常利用 $P$ 点附近重力加速度反常现象, 已知引力常量为 $G$ 。 设球形区域体积为 $V$, 球心深度为 $d$ ( $d$ 远小于地球半径), $\overline{P Q}=x$, 求: 球形区域内黄金矿石在 $Q$ 点产生的加速度大小; [图1] 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

EX

Astronomy

ZH

multi-modal

Astronomy_274

如图所示, 有 $a 、 b 、 c 、 d$ 四颗卫星, $a$ 未发射在地球赤道上随地球一起转动, $b$ 为近地轨道卫星, $c$ 为地球同步卫星, $d$ 为高空探测卫星, 所有卫星的运动均视为匀速圆周运动，重力加速度为 $g$, 则下列关于四颗卫星的说法正确的是（） [图1] A: $a$ 卫星的向心加速度等于重力加速度 $g$ B: $b$ 卫星与地心连线在单位时间扫过的面积等于 $c$ 卫星与地心连线在单位时间扫过的面积 C: $b 、 c$ 卫星轨道半径的三次方与周期平方之比相等 D: $a$ 卫星的运行周期大于 $d$ 卫星的运行周期

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 如图所示, 有 $a 、 b 、 c 、 d$ 四颗卫星, $a$ 未发射在地球赤道上随地球一起转动, $b$ 为近地轨道卫星, $c$ 为地球同步卫星, $d$ 为高空探测卫星, 所有卫星的运动均视为匀速圆周运动，重力加速度为 $g$, 则下列关于四颗卫星的说法正确的是（） [图1] A: $a$ 卫星的向心加速度等于重力加速度 $g$ B: $b$ 卫星与地心连线在单位时间扫过的面积等于 $c$ 卫星与地心连线在单位时间扫过的面积 C: $b 、 c$ 卫星轨道半径的三次方与周期平方之比相等 D: $a$ 卫星的运行周期大于 $d$ 卫星的运行周期 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

multi-modal

Astronomy_661

宇航员飞到一个被稠密气体包围的某行星上进行科学探索。他站在该行星表面, 从静止释放一个质量为 $m$ 的物体, 由于气体阻力的作用, 其加速度 $a$ 随下落位移 $x$ 变化的关系图像如图所示。已知该星球半径为 $R$, 万有引力常量为 $G$ 。下列说法正确的是 $(\quad)$ [图1] A: 该行星的平均密度为 $\frac{3 a_{0}}{4 \pi G R}$ B: 该行星的第一宇宙速度为 $\sqrt{a_{0} R}$ C: 卫星在距该行星表面高 $h$ 处的圆轨道上运行的周期为 $\frac{4 \pi}{R} \sqrt{\frac{(R+h)^{3}}{a_{0}}}$ D: 从释放到速度刚达到最大的过程中, 物体克服阻力做功 $\frac{m a_{0} x_{0}}{2}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 宇航员飞到一个被稠密气体包围的某行星上进行科学探索。他站在该行星表面, 从静止释放一个质量为 $m$ 的物体, 由于气体阻力的作用, 其加速度 $a$ 随下落位移 $x$ 变化的关系图像如图所示。已知该星球半径为 $R$, 万有引力常量为 $G$ 。下列说法正确的是 $(\quad)$ [图1] A: 该行星的平均密度为 $\frac{3 a_{0}}{4 \pi G R}$ B: 该行星的第一宇宙速度为 $\sqrt{a_{0} R}$ C: 卫星在距该行星表面高 $h$ 处的圆轨道上运行的周期为 $\frac{4 \pi}{R} \sqrt{\frac{(R+h)^{3}}{a_{0}}}$ D: 从释放到速度刚达到最大的过程中, 物体克服阻力做功 $\frac{m a_{0} x_{0}}{2}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_189

宇宙中有一孤立星系, 中心天体周围有三颗行星, 如图所示。中心天体质量远大于行星质量, 不考虑行星之间的万有引力, 三颗行星的运动轨道中, 有两个为圆轨道, 半径分别为 $r_{1} 、 r_{3}$, 一个为粗圆轨道, 半长轴为 $a, a=r_{3}$ 。在 $\Delta t$ 时间内, 行星II、行星III 与中心天体连线扫过的面积分别为 $S_{2} 、 S_{3}$ 。行星 I 的速率为 $v_{1}$, 行星 II 在 $B$ 点的速率为 $v_{2 \mathrm{~B}}$, 行星 II在 $E$ 点的速率为 $v_{2 \mathrm{E}}$, 行星III的速率为 $v_{3}$, 下列说法正确的是 ( ) [图1] A: $S_{2}<S_{3}$ B: 行星II与行星III的运行周期不相等 C: 行星II与行星III在 $P$ 点时的加速度大小不相等 D: $v_{2 \mathrm{E}}<v_{3}<v_{1}<v_{2 \mathrm{~B}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 宇宙中有一孤立星系, 中心天体周围有三颗行星, 如图所示。中心天体质量远大于行星质量, 不考虑行星之间的万有引力, 三颗行星的运动轨道中, 有两个为圆轨道, 半径分别为 $r_{1} 、 r_{3}$, 一个为粗圆轨道, 半长轴为 $a, a=r_{3}$ 。在 $\Delta t$ 时间内, 行星II、行星III 与中心天体连线扫过的面积分别为 $S_{2} 、 S_{3}$ 。行星 I 的速率为 $v_{1}$, 行星 II 在 $B$ 点的速率为 $v_{2 \mathrm{~B}}$, 行星 II在 $E$ 点的速率为 $v_{2 \mathrm{E}}$, 行星III的速率为 $v_{3}$, 下列说法正确的是 ( ) [图1] A: $S_{2}<S_{3}$ B: 行星II与行星III的运行周期不相等 C: 行星II与行星III在 $P$ 点时的加速度大小不相等 D: $v_{2 \mathrm{E}}<v_{3}<v_{1}<v_{2 \mathrm{~B}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_1020

An observer in the UK sees a full moon in the constellation of Capricorn. What time of year is it? A: Spring B: Summer C: Autumn D: Winter

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: An observer in the UK sees a full moon in the constellation of Capricorn. What time of year is it? A: Spring B: Summer C: Autumn D: Winter You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

SC

Astronomy

EN

text-only

Astronomy_959

Forces of Nature In the BBC programme Forces of Nature, Brian Cox uses a Eurofighter Typhoon to try and overtake the spin of the Earth such that the setting Sun appears to rise instead. [figure1] By considering the circumference of the Earth at each point, and the length of a day, what speed in the air would the Eurofighter need to achieve for the Sun to appear stationary if it took off from: i. The equator ii. Oxford (which has a latitude of about $52^{\circ}$ ) [You can ignore the altitude of the fighter jet]

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. problem: Forces of Nature In the BBC programme Forces of Nature, Brian Cox uses a Eurofighter Typhoon to try and overtake the spin of the Earth such that the setting Sun appears to rise instead. [figure1] By considering the circumference of the Earth at each point, and the length of a day, what speed in the air would the Eurofighter need to achieve for the Sun to appear stationary if it took off from: i. The equator ii. Oxford (which has a latitude of about $52^{\circ}$ ) [You can ignore the altitude of the fighter jet] All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of m/s, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

multi-modal

Astronomy_875

Orion is observing the sky with two telescopes that he just made. Orion wrote down that the first telescope has a primary mirror with focal length $F_{p}=2 \mathrm{~m}$ and an eye piece with focal length $F_{e}=30 \mathrm{~mm}$. However, he does not know the specifications of his second telescope. Given that the full-field image on the left was taken by the first telescope, and the full-field image on the right was taken by the second telescope, which of the following choices could be the specifications of the second telescope? [figure1] [figure2] A: $F_{p}=1 \mathrm{~m} F_{e}=15 \mathrm{~mm}$ B: $F_{p}=1 \mathrm{~m} F_{e}=30 \mathrm{~mm}$ C: $F_{p}=1 \mathrm{~m} F_{e}=90 \mathrm{~mm}$ D: $F_{p}=6 \mathrm{~m} F_{e}=30 \mathrm{~mm}$ E: $F_{p}=6 \mathrm{~m} F_{e}=15 \mathrm{~mm}$

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: Orion is observing the sky with two telescopes that he just made. Orion wrote down that the first telescope has a primary mirror with focal length $F_{p}=2 \mathrm{~m}$ and an eye piece with focal length $F_{e}=30 \mathrm{~mm}$. However, he does not know the specifications of his second telescope. Given that the full-field image on the left was taken by the first telescope, and the full-field image on the right was taken by the second telescope, which of the following choices could be the specifications of the second telescope? [figure1] [figure2] A: $F_{p}=1 \mathrm{~m} F_{e}=15 \mathrm{~mm}$ B: $F_{p}=1 \mathrm{~m} F_{e}=30 \mathrm{~mm}$ C: $F_{p}=1 \mathrm{~m} F_{e}=90 \mathrm{~mm}$ D: $F_{p}=6 \mathrm{~m} F_{e}=30 \mathrm{~mm}$ E: $F_{p}=6 \mathrm{~m} F_{e}=15 \mathrm{~mm}$ You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D, E].

SC

Astronomy

EN

multi-modal

Astronomy_70

宇航员在地球表面一斜坡上 $P$ 点, 沿水平方向以初速度 $v_{0}$ 抛出一个小球, 测得小球经时间 $t$ 落到斜坡另一点 $Q$ 上现宇航员站在某质量分布均匀的星球表面相同的斜坡上 $P$点, 沿水平方向以相同的初速度 $v_{0}$ 抛出一个小球, 小球落在 $P Q$ 的中点. 已知该星球的半径为 $R$, 地球表面重力加速度为 $g$, 引力常量为 $G$, 球的体积公式是 $V=\frac{4}{3} \pi R^{3}$ 。求: 该星球表面的重力加速度 $g_{0}$; [图1]

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 宇航员在地球表面一斜坡上 $P$ 点, 沿水平方向以初速度 $v_{0}$ 抛出一个小球, 测得小球经时间 $t$ 落到斜坡另一点 $Q$ 上现宇航员站在某质量分布均匀的星球表面相同的斜坡上 $P$点, 沿水平方向以相同的初速度 $v_{0}$ 抛出一个小球, 小球落在 $P Q$ 的中点. 已知该星球的半径为 $R$, 地球表面重力加速度为 $g$, 引力常量为 $G$, 球的体积公式是 $V=\frac{4}{3} \pi R^{3}$ 。求: 该星球表面的重力加速度 $g_{0}$; [图1] 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

EX

Astronomy

ZH

multi-modal

Astronomy_515

宇宙飞船以周期 $T$ 绕地球做圆周运动时, 由于地球遮挡阳光, 会经历“日全食”过程(宇航员看不见太阳), 如图所示, 已知地球的半径为 $R$, 地球质量为 $M$, 引力常量为 $G$,地球自转周期为 $T_{0}$, 太阳光可看作平行光, 飞船上的宇航员在 $A$ 点测出对地球的张角为 $a$ ，则以下判断正确的是( ) [图1] A: 飞船绕地球运动的线速度为 $\frac{2 \pi R}{T \sin \left(\frac{\alpha}{2}\right)}$ B: 一个天内飞船经历“日全食”的次数为 $\frac{2 T_{0}}{T}$ C: 飞船每次“日全食”过程的时间为 $\frac{\alpha T_{0}}{2 \pi}$ D: 飞船周期为 $T=\frac{2 \pi R}{\sin \left(\frac{\alpha}{2}\right)} \sqrt{\frac{R}{G M \sin \left(\frac{\alpha}{2}\right)}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 宇宙飞船以周期 $T$ 绕地球做圆周运动时, 由于地球遮挡阳光, 会经历“日全食”过程(宇航员看不见太阳), 如图所示, 已知地球的半径为 $R$, 地球质量为 $M$, 引力常量为 $G$,地球自转周期为 $T_{0}$, 太阳光可看作平行光, 飞船上的宇航员在 $A$ 点测出对地球的张角为 $a$ ，则以下判断正确的是( ) [图1] A: 飞船绕地球运动的线速度为 $\frac{2 \pi R}{T \sin \left(\frac{\alpha}{2}\right)}$ B: 一个天内飞船经历“日全食”的次数为 $\frac{2 T_{0}}{T}$ C: 飞船每次“日全食”过程的时间为 $\frac{\alpha T_{0}}{2 \pi}$ D: 飞船周期为 $T=\frac{2 \pi R}{\sin \left(\frac{\alpha}{2}\right)} \sqrt{\frac{R}{G M \sin \left(\frac{\alpha}{2}\right)}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_97

假设地球的两颗卫星 $\mathrm{A} 、 \mathrm{~B}$ 的运行方向相同周期之比为 $8: 1$, 其中 $\mathrm{B}$ 为近地卫星, 某时刻两卫星相距最近, 已知地球半径为 $R$, 地球表面的重力加速度为 $g$, 则 ( ) [图1] A: 卫星 $\mathrm{A} 、 \mathrm{~B}$ 运行的加速度大小之比 $\frac{a_{A}}{a_{B}}=\frac{1}{4}$ B: 卫星 $\mathrm{A} 、 \mathrm{~B}$ 运行的半径之比 $\frac{r_{A}}{r_{B}}=\frac{2}{1}$ C: 卫星 A、B 运行的线速度大小之比 $\frac{v_{A}}{v_{B}}=\frac{1}{2}$ D: 经过时间 $t=\frac{16 \pi}{7} \sqrt{\frac{R}{g}}$, 卫星 $\mathrm{A} 、 \mathrm{~B}$ 再次相距最近

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 假设地球的两颗卫星 $\mathrm{A} 、 \mathrm{~B}$ 的运行方向相同周期之比为 $8: 1$, 其中 $\mathrm{B}$ 为近地卫星, 某时刻两卫星相距最近, 已知地球半径为 $R$, 地球表面的重力加速度为 $g$, 则 ( ) [图1] A: 卫星 $\mathrm{A} 、 \mathrm{~B}$ 运行的加速度大小之比 $\frac{a_{A}}{a_{B}}=\frac{1}{4}$ B: 卫星 $\mathrm{A} 、 \mathrm{~B}$ 运行的半径之比 $\frac{r_{A}}{r_{B}}=\frac{2}{1}$ C: 卫星 A、B 运行的线速度大小之比 $\frac{v_{A}}{v_{B}}=\frac{1}{2}$ D: 经过时间 $t=\frac{16 \pi}{7} \sqrt{\frac{R}{g}}$, 卫星 $\mathrm{A} 、 \mathrm{~B}$ 再次相距最近 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_1131

It is often said that the Sun rises in the East and sets in the West, however this is only true twice a year at the equinoxes. In the Northern hemisphere, the Sun will rise northwards of East on the June solstice, and southwards of East on the December solstice; this is directly tied in with the varying length of day too, since the Sun either has a greater or shorter distance to travel across the sky (see Figure 1). [figure1] Figure 1: Left: The path of the Sun across the sky during the equinoxes and solstices, as viewed by an observer in the Northern hemisphere at a latitude of $\sim 40^{\circ}$. Credit: Daniel V. Schroeder / Weber State University. Right: The same idea but viewed from Iceland at a latitude of $65^{\circ}$, where by being so close to the Artic circle the day length can get close to 24 hours in June and almost no daylight in December. Credit: Kristn Bjarnadttir / University of Iceland. During the equinox, the Sun travels along the projection of the Earth's equator. In this question, we will assume a circular orbit for the Earth, and all angles will be calculated in degrees. A simple model for the vertical angle between the Sun and the horizon (known at the altitude), $h$, as a function of the bearing on the horizon, $A$ (measured clockwise from North, also called the azimuth), the latitude of the observer, $\phi$ (positive in Northern hemisphere, negative in Southern hemisphere), and the vertical angle of the Sun relative to the celestial equator (known as the solar declination), $\delta$, is given as: $$ h=-\left(90^{\circ}-\phi\right) \cos (A)+\delta $$ The solar declination can be considered to vary sinusoidally over the year, going from a maximum of $\delta=+23.44^{\circ}$ at the June solstice (roughly $21^{\text {st }}$ June) to a minimum of $\delta=-23.44^{\circ}$ on the December solstice (roughly $21^{\text {st }}$ December). It can be shown using spherical trigonometry that the precise model connecting $\delta, h, \phi$ and $A$ is: $$ \sin (\delta)=\sin (h) \sin (\phi)+\cos (h) \cos (\phi) \cos (A) . $$ Using the precise model, the path of the Sun across the sky forms a shape that is not quite the cosine shape of the simple model, and is shown in Figure 2. [figure2] Figure 2: The altitude of the Sun as a function of bearing during the equinoxes and solstices, as viewed by an observer at a latitude of $+56^{\circ}$. Whilst it resembles the cosine shape of the simple model well at this latitude, there are small deviations. Credit: Wikipedia. By using further spherical trigonometry, we can derive a second helpful equation in the precise model: $$ \sin (h)=\sin (\phi) \sin (\delta)+\cos (\phi) \cos (\delta) \cos (H) $$ Here, $H$ is the solar hour angle, which measures the angle between the Sun and solar noon as measured along the projection of the Earth's equator on the sky. Conventionally, $H=0^{\circ}$ at solar noon, is negative before solar noon, and is positive afterwards. Since the sun's hour angle increases at an approximately constant rate due to the rotation of the Earth, we can convert this angle into a time using the conversion $360^{\circ}=24^{\mathrm{h}}$.a. Consider an observer in Oxford $\left(\phi=+51.8^{\circ}\right)$ on the June solstice. i. Calculate the bearing of sunrise. Take the Sun to be a point source and ignore any atmospheric effects.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: It is often said that the Sun rises in the East and sets in the West, however this is only true twice a year at the equinoxes. In the Northern hemisphere, the Sun will rise northwards of East on the June solstice, and southwards of East on the December solstice; this is directly tied in with the varying length of day too, since the Sun either has a greater or shorter distance to travel across the sky (see Figure 1). [figure1] Figure 1: Left: The path of the Sun across the sky during the equinoxes and solstices, as viewed by an observer in the Northern hemisphere at a latitude of $\sim 40^{\circ}$. Credit: Daniel V. Schroeder / Weber State University. Right: The same idea but viewed from Iceland at a latitude of $65^{\circ}$, where by being so close to the Artic circle the day length can get close to 24 hours in June and almost no daylight in December. Credit: Kristn Bjarnadttir / University of Iceland. During the equinox, the Sun travels along the projection of the Earth's equator. In this question, we will assume a circular orbit for the Earth, and all angles will be calculated in degrees. A simple model for the vertical angle between the Sun and the horizon (known at the altitude), $h$, as a function of the bearing on the horizon, $A$ (measured clockwise from North, also called the azimuth), the latitude of the observer, $\phi$ (positive in Northern hemisphere, negative in Southern hemisphere), and the vertical angle of the Sun relative to the celestial equator (known as the solar declination), $\delta$, is given as: $$ h=-\left(90^{\circ}-\phi\right) \cos (A)+\delta $$ The solar declination can be considered to vary sinusoidally over the year, going from a maximum of $\delta=+23.44^{\circ}$ at the June solstice (roughly $21^{\text {st }}$ June) to a minimum of $\delta=-23.44^{\circ}$ on the December solstice (roughly $21^{\text {st }}$ December). It can be shown using spherical trigonometry that the precise model connecting $\delta, h, \phi$ and $A$ is: $$ \sin (\delta)=\sin (h) \sin (\phi)+\cos (h) \cos (\phi) \cos (A) . $$ Using the precise model, the path of the Sun across the sky forms a shape that is not quite the cosine shape of the simple model, and is shown in Figure 2. [figure2] Figure 2: The altitude of the Sun as a function of bearing during the equinoxes and solstices, as viewed by an observer at a latitude of $+56^{\circ}$. Whilst it resembles the cosine shape of the simple model well at this latitude, there are small deviations. Credit: Wikipedia. By using further spherical trigonometry, we can derive a second helpful equation in the precise model: $$ \sin (h)=\sin (\phi) \sin (\delta)+\cos (\phi) \cos (\delta) \cos (H) $$ Here, $H$ is the solar hour angle, which measures the angle between the Sun and solar noon as measured along the projection of the Earth's equator on the sky. Conventionally, $H=0^{\circ}$ at solar noon, is negative before solar noon, and is positive afterwards. Since the sun's hour angle increases at an approximately constant rate due to the rotation of the Earth, we can convert this angle into a time using the conversion $360^{\circ}=24^{\mathrm{h}}$. problem: a. Consider an observer in Oxford $\left(\phi=+51.8^{\circ}\right)$ on the June solstice. i. Calculate the bearing of sunrise. Take the Sun to be a point source and ignore any atmospheric effects. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of ^{\circ}, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

multi-modal

Astronomy_143

019 年 1 月 3 号 “嫦娥 4 号”探测器实现人类首次月球背面着陆, 并开展巡视探测。因月球没有大气，无法通过降落企减速着陆，必须通过引擎喷射来实现减速。如图所示 为“嫦娥 4 号”探测器降落月球表面过程的简化模型。质量 $m$ 的探测器沿半径为 $r$ 的圆轨道 $\mathrm{I}$ 绕月运动。为使探测器安全着陆, 首先在 $\mathrm{P}$ 点沿轨道切线方向向前以速度 $u$ 喷射质量为 $\triangle m$ 的物体, 从而使探测器由 $\mathrm{P}$ 点沿椭圆轨道 II 转至 $\mathrm{Q}$ 点（椭圆轨道与月球在 $\mathrm{Q}$点相切）时恰好到达月球表面附近, 再次向前喷射减速着陆。已知月球质量为 $\mathrm{M}$ 、半径为 $R$ 。万有引力常量为 $\mathrm{G}$ 。则下列说法正确的是 ( ) [图1] A: 探测器喷射物体前在圆周轨道 I 上运行时的周期为 $2 \pi \sqrt{\frac{r^{3}}{G M}}$ B: 在 $\mathrm{P}$ 点探测器喷射物体后速度大小变为 $\frac{(m-\Delta m) u}{m}$ C: 减速降落过程, 从 P 点沿轨道 II 运行到月球表面所经历的时间为 $\frac{\pi}{2} \sqrt{\frac{(R+r)^{3}}{G M}}$ D: 月球表面重力加速度的大小为 $\frac{G M}{R^{2}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 019 年 1 月 3 号 “嫦娥 4 号”探测器实现人类首次月球背面着陆, 并开展巡视探测。因月球没有大气，无法通过降落企减速着陆，必须通过引擎喷射来实现减速。如图所示 为“嫦娥 4 号”探测器降落月球表面过程的简化模型。质量 $m$ 的探测器沿半径为 $r$ 的圆轨道 $\mathrm{I}$ 绕月运动。为使探测器安全着陆, 首先在 $\mathrm{P}$ 点沿轨道切线方向向前以速度 $u$ 喷射质量为 $\triangle m$ 的物体, 从而使探测器由 $\mathrm{P}$ 点沿椭圆轨道 II 转至 $\mathrm{Q}$ 点（椭圆轨道与月球在 $\mathrm{Q}$点相切）时恰好到达月球表面附近, 再次向前喷射减速着陆。已知月球质量为 $\mathrm{M}$ 、半径为 $R$ 。万有引力常量为 $\mathrm{G}$ 。则下列说法正确的是 ( ) [图1] A: 探测器喷射物体前在圆周轨道 I 上运行时的周期为 $2 \pi \sqrt{\frac{r^{3}}{G M}}$ B: 在 $\mathrm{P}$ 点探测器喷射物体后速度大小变为 $\frac{(m-\Delta m) u}{m}$ C: 减速降落过程, 从 P 点沿轨道 II 运行到月球表面所经历的时间为 $\frac{\pi}{2} \sqrt{\frac{(R+r)^{3}}{G M}}$ D: 月球表面重力加速度的大小为 $\frac{G M}{R^{2}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_766

The planet's albedo is the fraction of incident light ... A: reflected by the planet's surface. B: absorbed by the planet's surface. C: blocked by the planet's surface. D: emitted by the planet's surface.

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: The planet's albedo is the fraction of incident light ... A: reflected by the planet's surface. B: absorbed by the planet's surface. C: blocked by the planet's surface. D: emitted by the planet's surface. You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

SC

Astronomy

EN

text-only

Astronomy_785

Why is it hard to observe the universe with infrared radiation from the Earth's surface? A: Harmful to humans B: Blocked by the atmosphere C: Reflected by the atmosphere D: Distorted by the atmosphere

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: Why is it hard to observe the universe with infrared radiation from the Earth's surface? A: Harmful to humans B: Blocked by the atmosphere C: Reflected by the atmosphere D: Distorted by the atmosphere You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

SC

Astronomy

EN

text-only

Astronomy_1091

The Event Horizon Telescope (EHT) is a project to use many widely-spaced radio telescopes as a Very Long Baseline Interferometer (VBLI) to create a virtual telescope as big as the Earth. This extraordinary size allows sufficient angular resolution to be able to image the space close to the event horizon of a super massive black hole (SMBH), and provide an opportunity to test the predictions of Einstein's theory of General Relativity (GR) in a very strong gravitational field. In April 2017 the EHT collaboration managed to co-ordinate time on all of the telescopes in the array so that they could observe the SMBH (called M87*) at the centre of the Virgo galaxy, M87, and they plan to also image the SMBH at the centre of our galaxy (called Sgr A*). [figure1] Figure 3: Left: The locations of all the telescopes used during the April 2017 observing run. The solid lines correspond to baselines used for observing M87, whilst the dashed lines were the baselines used for the calibration source. Credit: EHT Collaboration. Right: A simulated model of what the region near an SMBH could look like, modelled at much higher resolution than the EHT can achieve. The light comes from the accretion disc, but the paths of the photons are bent into a characteristic shape by the extreme gravity, leading to a 'shadow' in middle of the disc - this is what the EHT is trying to image. The left side of the image is brighter than the right side as light emitted from a substance moving towards an observer is brighter than that of one moving away. Credit: Hotaka Shiokawa. Some data about the locations of the eight telescopes in the array are given below in 3-D cartesian geocentric coordinates with $X$ pointing to the Greenwich meridian, $Y$ pointing $90^{\circ}$ away in the equatorial plane (eastern longitudes have positive $Y$ ), and positive $Z$ pointing in the direction of the North Pole. This is a left-handed coordinate system. | Facility | Location | $X(\mathrm{~m})$ | $Y(\mathrm{~m})$ | $Z(\mathrm{~m})$ | | :--- | :--- | :---: | :---: | :---: | | ALMA | Chile | 2225061.3 | -5440061.7 | -2481681.2 | | APEX | Chile | 2225039.5 | -5441197.6 | -2479303.4 | | JCMT | Hawaii, USA | -5464584.7 | -2493001.2 | 2150654.0 | | LMT | Mexico | -768715.6 | -5988507.1 | 2063354.9 | | PV | Spain | 5088967.8 | -301681.2 | 3825012.2 | | SMA | Hawaii, USA | -5464555.5 | -2492928.0 | 2150797.2 | | SMT | Arizona, USA | -1828796.2 | -5054406.8 | 3427865.2 | | SPT | Antarctica | 809.8 | -816.9 | -6359568.7 | The minimum angle, $\theta_{\min }$ (in radians) that can be resolved by a VLBI array is given by the equation $$ \theta_{\min }=\frac{\lambda_{\mathrm{obs}}}{d_{\max }}, $$ where $\lambda_{\text {obs }}$ is the observing wavelength and $d_{\max }$ is the longest straight line distance between two telescopes used (called the baseline), assumed perpendicular to the line of sight during the observation. An important length scale when discussing black holes is the gravitational radius, $r_{g}=\frac{G M}{c^{2}}$, where $G$ is the gravitational constant, $M$ is the mass of the black hole and $c$ is the speed of light. The familiar event horizon of a non-rotating black hole is called the Schwartzschild radius, $r_{S} \equiv 2 r_{g}$, however this is not what the EHT is able to observe - instead the closest it can see to a black hole is called the photon sphere, where photons orbit in the black hole in unstable circular orbits. On top of this the image of the black hole is gravitationally lensed by the black hole itself magnifying the apparent radius of the photon sphere to be between $(2 \sqrt{3+2 \sqrt{2}}) r_{g}$ and $(3 \sqrt{3}) r_{g}$, determined by spin and inclination; the latter corresponds to a perfectly spherical non-spinning black hole. The area within this lensed image will appear almost black and is the 'shadow' the EHT is looking for. [figure2] Figure 4: Four nights of data were taken for M87* during the observing window of the EHT, and whilst the diameter of the disk stayed relatively constant the location of bright spots moved, possibly indicating gas that is orbiting the black hole. Credit: EHT Collaboration. The EHT observed M87* on four separate occasions during the observing window (see Fig 4), and the team saw that some of the bright spots changed in that time, suggesting they may be associated with orbiting gas close to the black hole. The Innermost Stable Circular Orbit (ISCO) is the equivalent of the photon sphere but for particles with mass (and is also stable). The total conserved energy of a circular orbit close to a non-spinning black hole is given by $$ E=m c^{2}\left(\frac{1-\frac{2 r_{g}}{r}}{\sqrt{1-\frac{3 r_{g}}{r}}}\right) $$ and the radius of the ISCO, $r_{\mathrm{ISCO}}$, is the value of $r$ for which $E$ is minimised. We expect that most black holes are in fact spinning (since most stars are spinning) and the spin of a black hole is quantified with the spin parameter $a \equiv J / J_{\max }$ where $J$ is the angular momentum of the black hole and $J_{\max }=G M^{2} / c$ is the maximum possible angular momentum it can have. The value of $a$ varies from $-1 \leq a \leq 1$, where negative spins correspond to the black hole rotating in the opposite direction to its accretion disk, and positive spins in the same direction. If $a=1$ then $r_{\text {ISCO }}=r_{g}$, whilst if $a=-1$ then $r_{\text {ISCO }}=9 r_{g}$. The angular velocity of a particle in the ISCO is given by $$ \omega^{2}=\frac{G M}{\left(r_{\text {ISCO }}^{3 / 2}+a r_{g}^{3 / 2}\right)^{2}} $$ [figure3] Figure 5: Due to the curvature of spacetime, the real distance travelled by a particle moving from the ISCO to the photon sphere (indicated with the solid red arrow) is longer than you would get purely from subtracting the radial co-ordinates of those orbits (indicated with the dashed blue arrow), which would be valid for a flat spacetime. Relations between these distances are not to scale in this diagram. Credit: Modified from Bardeen et al. (1972). The spacetime near a black hole is curved, as described by the equations of GR. This means that the distance between two points can be substantially different to the distance you would expect if spacetime was flat. GR tells us that the proper distance travelled by a particle moving from radius $r_{1}$ to radius $r_{2}$ around a black hole of mass $M$ (with $r_{1}>r_{2}$ ) is given by $$ \Delta l=\int_{r_{2}}^{r_{1}}\left(1-\frac{2 r_{g}}{r}\right)^{-1 / 2} \mathrm{~d} r $$f. A particle close to M87* moves directly from risco $^{\text {to }} r_{\text {ph }}$ (and subsequently into the black hole). What is the extra distance travelled by it due to the curvature of spacetime, as described in Fig 5 ? Give your answer in au, and assume M87* is non-spinning.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is an expression. Here is some context information for this question, which might assist you in solving it: The Event Horizon Telescope (EHT) is a project to use many widely-spaced radio telescopes as a Very Long Baseline Interferometer (VBLI) to create a virtual telescope as big as the Earth. This extraordinary size allows sufficient angular resolution to be able to image the space close to the event horizon of a super massive black hole (SMBH), and provide an opportunity to test the predictions of Einstein's theory of General Relativity (GR) in a very strong gravitational field. In April 2017 the EHT collaboration managed to co-ordinate time on all of the telescopes in the array so that they could observe the SMBH (called M87*) at the centre of the Virgo galaxy, M87, and they plan to also image the SMBH at the centre of our galaxy (called Sgr A*). [figure1] Figure 3: Left: The locations of all the telescopes used during the April 2017 observing run. The solid lines correspond to baselines used for observing M87, whilst the dashed lines were the baselines used for the calibration source. Credit: EHT Collaboration. Right: A simulated model of what the region near an SMBH could look like, modelled at much higher resolution than the EHT can achieve. The light comes from the accretion disc, but the paths of the photons are bent into a characteristic shape by the extreme gravity, leading to a 'shadow' in middle of the disc - this is what the EHT is trying to image. The left side of the image is brighter than the right side as light emitted from a substance moving towards an observer is brighter than that of one moving away. Credit: Hotaka Shiokawa. Some data about the locations of the eight telescopes in the array are given below in 3-D cartesian geocentric coordinates with $X$ pointing to the Greenwich meridian, $Y$ pointing $90^{\circ}$ away in the equatorial plane (eastern longitudes have positive $Y$ ), and positive $Z$ pointing in the direction of the North Pole. This is a left-handed coordinate system. | Facility | Location | $X(\mathrm{~m})$ | $Y(\mathrm{~m})$ | $Z(\mathrm{~m})$ | | :--- | :--- | :---: | :---: | :---: | | ALMA | Chile | 2225061.3 | -5440061.7 | -2481681.2 | | APEX | Chile | 2225039.5 | -5441197.6 | -2479303.4 | | JCMT | Hawaii, USA | -5464584.7 | -2493001.2 | 2150654.0 | | LMT | Mexico | -768715.6 | -5988507.1 | 2063354.9 | | PV | Spain | 5088967.8 | -301681.2 | 3825012.2 | | SMA | Hawaii, USA | -5464555.5 | -2492928.0 | 2150797.2 | | SMT | Arizona, USA | -1828796.2 | -5054406.8 | 3427865.2 | | SPT | Antarctica | 809.8 | -816.9 | -6359568.7 | The minimum angle, $\theta_{\min }$ (in radians) that can be resolved by a VLBI array is given by the equation $$ \theta_{\min }=\frac{\lambda_{\mathrm{obs}}}{d_{\max }}, $$ where $\lambda_{\text {obs }}$ is the observing wavelength and $d_{\max }$ is the longest straight line distance between two telescopes used (called the baseline), assumed perpendicular to the line of sight during the observation. An important length scale when discussing black holes is the gravitational radius, $r_{g}=\frac{G M}{c^{2}}$, where $G$ is the gravitational constant, $M$ is the mass of the black hole and $c$ is the speed of light. The familiar event horizon of a non-rotating black hole is called the Schwartzschild radius, $r_{S} \equiv 2 r_{g}$, however this is not what the EHT is able to observe - instead the closest it can see to a black hole is called the photon sphere, where photons orbit in the black hole in unstable circular orbits. On top of this the image of the black hole is gravitationally lensed by the black hole itself magnifying the apparent radius of the photon sphere to be between $(2 \sqrt{3+2 \sqrt{2}}) r_{g}$ and $(3 \sqrt{3}) r_{g}$, determined by spin and inclination; the latter corresponds to a perfectly spherical non-spinning black hole. The area within this lensed image will appear almost black and is the 'shadow' the EHT is looking for. [figure2] Figure 4: Four nights of data were taken for M87* during the observing window of the EHT, and whilst the diameter of the disk stayed relatively constant the location of bright spots moved, possibly indicating gas that is orbiting the black hole. Credit: EHT Collaboration. The EHT observed M87* on four separate occasions during the observing window (see Fig 4), and the team saw that some of the bright spots changed in that time, suggesting they may be associated with orbiting gas close to the black hole. The Innermost Stable Circular Orbit (ISCO) is the equivalent of the photon sphere but for particles with mass (and is also stable). The total conserved energy of a circular orbit close to a non-spinning black hole is given by $$ E=m c^{2}\left(\frac{1-\frac{2 r_{g}}{r}}{\sqrt{1-\frac{3 r_{g}}{r}}}\right) $$ and the radius of the ISCO, $r_{\mathrm{ISCO}}$, is the value of $r$ for which $E$ is minimised. We expect that most black holes are in fact spinning (since most stars are spinning) and the spin of a black hole is quantified with the spin parameter $a \equiv J / J_{\max }$ where $J$ is the angular momentum of the black hole and $J_{\max }=G M^{2} / c$ is the maximum possible angular momentum it can have. The value of $a$ varies from $-1 \leq a \leq 1$, where negative spins correspond to the black hole rotating in the opposite direction to its accretion disk, and positive spins in the same direction. If $a=1$ then $r_{\text {ISCO }}=r_{g}$, whilst if $a=-1$ then $r_{\text {ISCO }}=9 r_{g}$. The angular velocity of a particle in the ISCO is given by $$ \omega^{2}=\frac{G M}{\left(r_{\text {ISCO }}^{3 / 2}+a r_{g}^{3 / 2}\right)^{2}} $$ [figure3] Figure 5: Due to the curvature of spacetime, the real distance travelled by a particle moving from the ISCO to the photon sphere (indicated with the solid red arrow) is longer than you would get purely from subtracting the radial co-ordinates of those orbits (indicated with the dashed blue arrow), which would be valid for a flat spacetime. Relations between these distances are not to scale in this diagram. Credit: Modified from Bardeen et al. (1972). The spacetime near a black hole is curved, as described by the equations of GR. This means that the distance between two points can be substantially different to the distance you would expect if spacetime was flat. GR tells us that the proper distance travelled by a particle moving from radius $r_{1}$ to radius $r_{2}$ around a black hole of mass $M$ (with $r_{1}>r_{2}$ ) is given by $$ \Delta l=\int_{r_{2}}^{r_{1}}\left(1-\frac{2 r_{g}}{r}\right)^{-1 / 2} \mathrm{~d} r $$ problem: f. A particle close to M87* moves directly from risco $^{\text {to }} r_{\text {ph }}$ (and subsequently into the black hole). What is the extra distance travelled by it due to the curvature of spacetime, as described in Fig 5 ? Give your answer in au, and assume M87* is non-spinning. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is an expression without equals signs, e.g. ANSWER=\frac{1}{2} g t^2

EX

Astronomy

EN

multi-modal

Astronomy_53

某宇宙飞船在赤道所在平面内绕地球做匀速圆周运动, 假设地球赤道平面与其公转平面共面, 地球半径为 R. 日落后 3 小时时, 站在地球赤道上的小明, 刚好观察到头顶正上方的宇宙飞船正要进入地球阴影区，则 A: 宇宙飞船距地面高度为 $\sqrt{2} \mathrm{R}$ B: 在宇宙飞船中的宇航员观测地球, 其张角为 $90^{\circ}$ C: 宇航员绕地球一周经历的“夜晚”时间为 6 小时 D: 若宇宙飞船的周期为 $T$ ，则宇航员绕地球一周经历的“夜晚”时间为 $T / 4$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 某宇宙飞船在赤道所在平面内绕地球做匀速圆周运动, 假设地球赤道平面与其公转平面共面, 地球半径为 R. 日落后 3 小时时, 站在地球赤道上的小明, 刚好观察到头顶正上方的宇宙飞船正要进入地球阴影区，则 A: 宇宙飞船距地面高度为 $\sqrt{2} \mathrm{R}$ B: 在宇宙飞船中的宇航员观测地球, 其张角为 $90^{\circ}$ C: 宇航员绕地球一周经历的“夜晚”时间为 6 小时 D: 若宇宙飞船的周期为 $T$ ，则宇航员绕地球一周经历的“夜晚”时间为 $T / 4$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

text-only

Astronomy_968

In the UK we use the Gregorian calendar; it is a solar calendar so that a year corresponds to the time to orbit the Sun once, where 1 solar year is $\approx 365.25$ days. Several cultures use a lunar calendar, where each month is determined by the time it takes to go from New Moon to New Moon, and have a lunar year that is exactly 12 lunar months. An example of this is the Islamic calendar. Since the length of a lunar month (29.53 days) is a little shorter than the average month length in our solar calendar (see Figure 3), it means the start date of each month in the Islamic calendar is not tied to the seasons and gradually moves earlier in the solar year. [figure1] Figure 3: All the moon phases in May 2022, showing that the time measured from New Moon to New Moon (a lunar month) is shorter than a month in the Gregorian calendar. Credit: MoonConnection.com Given that the Islamic year $1429 \mathrm{AH}$ fell completely within the Gregorian year $2008 \mathrm{CE}$, calculate the Gregorian year in which the Islamic calendar started, and predict the Gregorian year when (at least in part of it) they will both have the same numerical value of year.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. problem: In the UK we use the Gregorian calendar; it is a solar calendar so that a year corresponds to the time to orbit the Sun once, where 1 solar year is $\approx 365.25$ days. Several cultures use a lunar calendar, where each month is determined by the time it takes to go from New Moon to New Moon, and have a lunar year that is exactly 12 lunar months. An example of this is the Islamic calendar. Since the length of a lunar month (29.53 days) is a little shorter than the average month length in our solar calendar (see Figure 3), it means the start date of each month in the Islamic calendar is not tied to the seasons and gradually moves earlier in the solar year. [figure1] Figure 3: All the moon phases in May 2022, showing that the time measured from New Moon to New Moon (a lunar month) is shorter than a month in the Gregorian calendar. Credit: MoonConnection.com Given that the Islamic year $1429 \mathrm{AH}$ fell completely within the Gregorian year $2008 \mathrm{CE}$, calculate the Gregorian year in which the Islamic calendar started, and predict the Gregorian year when (at least in part of it) they will both have the same numerical value of year. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value.

NV

Astronomy

EN

multi-modal

Astronomy_378

长征五号遥四运载火箭直接将我国首次执行火星探测任务的“天问一号””探测器送入地火转移轨道, 自此“天问一号”开启了奔向火星的旅程。如图所示为“天问一号”的运动轨迹图, 下列说法正确的是 ( ) [图1] A: 发射阶段的末速度已经超过了第二宇宙速度 B: 探测器沿不同轨道经过图中的 $A$ 点时的速度都相同 C: 探测器沿不同轨道经过图中的 $A$ 点时的加速度都相同 D: “天问一号”在火星着陆时, 发动机要向运动的反方向喷气

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 长征五号遥四运载火箭直接将我国首次执行火星探测任务的“天问一号””探测器送入地火转移轨道, 自此“天问一号”开启了奔向火星的旅程。如图所示为“天问一号”的运动轨迹图, 下列说法正确的是 ( ) [图1] A: 发射阶段的末速度已经超过了第二宇宙速度 B: 探测器沿不同轨道经过图中的 $A$ 点时的速度都相同 C: 探测器沿不同轨道经过图中的 $A$ 点时的加速度都相同 D: “天问一号”在火星着陆时, 发动机要向运动的反方向喷气 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_1048

GW170817 was the first gravitational wave event arising from a binary neutron star merger to have been detected by the LIGO \& Virgo experiments, and careful localization of the source meant that the electromagnetic counterpart was quickly found in galaxy NGC 4993 (see Figure 2). Such a combination of two completely separate branches of astronomical observation begins a new era of 'multi-messenger astronomy'. Since the gravitational waves allow an independent measurement of the distance to the host galaxy and the light allows an independent measurement of the recessional speed, this observation allows us to determine a new, independent value of the Hubble constant $H_{0}$. Another way of measuring distances to galaxies is to use the Fundamental Plane (FP) relation, which relies on the assumption there is a fairly tight relation between radius, surface brightness, and velocity dispersion for bulge-dominated galaxies, and is widely used for galaxies like NGC 4993. It can be described by the relation $$ \log \left(\frac{D}{(1+z)^{2}}\right)=-\log R_{e}+\alpha \log \sigma-\beta \log \left\langle I_{r}\right\rangle_{e}+\gamma $$ where $D$ is the distance in Mpc, $R_{e}$ is the effective radius measured in arcseconds, $\sigma$ is the velocity dispersion in $\mathrm{km} \mathrm{s}^{-1},\left\langle I_{r}\right\rangle_{e}$ is the mean intensity inside the effective radius measured in $L_{\odot} \mathrm{pc}^{-2}$, and $\gamma$ is the distance-dependent zero point of the relation. Calibrating the zero point to the Leo I galaxy group, the constants in the FP relation become $\alpha=1.24, \beta=0.82$, and $\gamma=2.194$. [figure1] Figure 2: Left: The GW170817 signal as measured by the LIGO and Virgo gravitational wave detectors, taken from Abbott $e t$ al. (2017). The normalized amplitude (or strain) is in units of $10^{-21}$. The signal is not visible in the Virgo data due to the direction of the source with respect to the detector's antenna pattern. Right: The optical counterpart of GW170817 in host galaxy NGC 4993, taken from Hjorth et al. (2017). By measuring the amplitude (called strain, $h$ ) and the frequency of the gravitational waves, $f_{\mathrm{GW}}$, one can determine the distance to the source without having to rely on 'standard candles' like Cepheid variables or Type Ia supernovae. For two masses, $m_{1}$ and $m_{2}$, orbiting the centre of mass with separation $a$ with orbital angular velocity $\omega$, then the dimensionless strain parameter $h$ is $$ h \simeq \frac{G}{c^{4}} \frac{1}{r} \mu a^{2} \omega^{2} $$ where $r$ is the luminosity distance, $c$ is the speed of light, $\mu=m_{1} m_{2} / M_{\text {tot }}$ is the reduced mass and $M_{\text {tot }}=m_{1}+m_{2}$ is the total mass.c. The Wolf-Rayet star WR7 is in the constellation of Canis Major and its strong winds are responsible for the nebula known as Thor's Helmet. The star has a mass of $16 M_{\odot}$, a radius of $1.41 R_{\odot}$ and a surface temperature of $112000 \mathrm{~K}$, with a measured $v_{\infty}$ of $1545 \mathrm{~km} \mathrm{~s}^{-1}$. iii. If the expansion is purely driven by the direct impact of the stellar winds, then the radius at time $t, R(t)$, can be related to $\dot{M}$ with the given formula. If $n_{0}=16 \mathrm{~cm}^{-3}$ and $m_{H}=1.67 \times 10^{-27}$ $\mathrm{kg}$, calculate the observed mass loss rate based upon the properties of the nebula. Compare it with the predicted one from earlier and comment on your answer.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: GW170817 was the first gravitational wave event arising from a binary neutron star merger to have been detected by the LIGO \& Virgo experiments, and careful localization of the source meant that the electromagnetic counterpart was quickly found in galaxy NGC 4993 (see Figure 2). Such a combination of two completely separate branches of astronomical observation begins a new era of 'multi-messenger astronomy'. Since the gravitational waves allow an independent measurement of the distance to the host galaxy and the light allows an independent measurement of the recessional speed, this observation allows us to determine a new, independent value of the Hubble constant $H_{0}$. Another way of measuring distances to galaxies is to use the Fundamental Plane (FP) relation, which relies on the assumption there is a fairly tight relation between radius, surface brightness, and velocity dispersion for bulge-dominated galaxies, and is widely used for galaxies like NGC 4993. It can be described by the relation $$ \log \left(\frac{D}{(1+z)^{2}}\right)=-\log R_{e}+\alpha \log \sigma-\beta \log \left\langle I_{r}\right\rangle_{e}+\gamma $$ where $D$ is the distance in Mpc, $R_{e}$ is the effective radius measured in arcseconds, $\sigma$ is the velocity dispersion in $\mathrm{km} \mathrm{s}^{-1},\left\langle I_{r}\right\rangle_{e}$ is the mean intensity inside the effective radius measured in $L_{\odot} \mathrm{pc}^{-2}$, and $\gamma$ is the distance-dependent zero point of the relation. Calibrating the zero point to the Leo I galaxy group, the constants in the FP relation become $\alpha=1.24, \beta=0.82$, and $\gamma=2.194$. [figure1] Figure 2: Left: The GW170817 signal as measured by the LIGO and Virgo gravitational wave detectors, taken from Abbott $e t$ al. (2017). The normalized amplitude (or strain) is in units of $10^{-21}$. The signal is not visible in the Virgo data due to the direction of the source with respect to the detector's antenna pattern. Right: The optical counterpart of GW170817 in host galaxy NGC 4993, taken from Hjorth et al. (2017). By measuring the amplitude (called strain, $h$ ) and the frequency of the gravitational waves, $f_{\mathrm{GW}}$, one can determine the distance to the source without having to rely on 'standard candles' like Cepheid variables or Type Ia supernovae. For two masses, $m_{1}$ and $m_{2}$, orbiting the centre of mass with separation $a$ with orbital angular velocity $\omega$, then the dimensionless strain parameter $h$ is $$ h \simeq \frac{G}{c^{4}} \frac{1}{r} \mu a^{2} \omega^{2} $$ where $r$ is the luminosity distance, $c$ is the speed of light, $\mu=m_{1} m_{2} / M_{\text {tot }}$ is the reduced mass and $M_{\text {tot }}=m_{1}+m_{2}$ is the total mass. problem: c. The Wolf-Rayet star WR7 is in the constellation of Canis Major and its strong winds are responsible for the nebula known as Thor's Helmet. The star has a mass of $16 M_{\odot}$, a radius of $1.41 R_{\odot}$ and a surface temperature of $112000 \mathrm{~K}$, with a measured $v_{\infty}$ of $1545 \mathrm{~km} \mathrm{~s}^{-1}$. iii. If the expansion is purely driven by the direct impact of the stellar winds, then the radius at time $t, R(t)$, can be related to $\dot{M}$ with the given formula. If $n_{0}=16 \mathrm{~cm}^{-3}$ and $m_{H}=1.67 \times 10^{-27}$ $\mathrm{kg}$, calculate the observed mass loss rate based upon the properties of the nebula. Compare it with the predicted one from earlier and comment on your answer. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of \mathrm{~J}, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

multi-modal

Astronomy_716

在暑期科幻电影《独行月球》中, 有大量的与航天知识有关的情景。已知月球的质量约为地球质量的 $\frac{1}{81}$, 月球半径约为地球半径的 $\frac{1}{4}$, 月球绕地球公转的周期约为 27 天。则下列说法正确的是 A: 从地球向月球发射的探月飞船的发射速度应大于 $11.2 \mathrm{~km} / \mathrm{s}$ B: 用同一速度坚直上抛一物体, 在月球表面的最大上升高度是地球表面的 $\frac{81}{16}$ 倍 C: 月球到地心的距离大约是地球同步卫星轨道半径的 9 倍 D: 月球的第一宇宙速度大于 $7.9 \mathrm{~km} / \mathrm{s}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 在暑期科幻电影《独行月球》中, 有大量的与航天知识有关的情景。已知月球的质量约为地球质量的 $\frac{1}{81}$, 月球半径约为地球半径的 $\frac{1}{4}$, 月球绕地球公转的周期约为 27 天。则下列说法正确的是 A: 从地球向月球发射的探月飞船的发射速度应大于 $11.2 \mathrm{~km} / \mathrm{s}$ B: 用同一速度坚直上抛一物体, 在月球表面的最大上升高度是地球表面的 $\frac{81}{16}$ 倍 C: 月球到地心的距离大约是地球同步卫星轨道半径的 9 倍 D: 月球的第一宇宙速度大于 $7.9 \mathrm{~km} / \mathrm{s}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

text-only

Astronomy_228

如图所示, 质量分别为 $M$ 和 $m$ 的两个星球 $\mathrm{A}$ 和 $\mathrm{B}$ (均视为质点) 在它们之间的引力作用下都绕 $O$ 点做匀速圆周运动, 星球 $\mathrm{A}$ 和 $\mathrm{B}$ 之间的距离为 $L$ 。已知星球 $\mathrm{A}$ 和 $\mathrm{B}$ 和 $O$三点始终共线, $\mathrm{A}$ 和 $\mathrm{B}$ 分别在 $O$ 点的两侧。引力常量为 $G$ 。 求星球 $\mathrm{B}$ 的线速度大小 $v$; [图1]

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 如图所示, 质量分别为 $M$ 和 $m$ 的两个星球 $\mathrm{A}$ 和 $\mathrm{B}$ (均视为质点) 在它们之间的引力作用下都绕 $O$ 点做匀速圆周运动, 星球 $\mathrm{A}$ 和 $\mathrm{B}$ 之间的距离为 $L$ 。已知星球 $\mathrm{A}$ 和 $\mathrm{B}$ 和 $O$三点始终共线, $\mathrm{A}$ 和 $\mathrm{B}$ 分别在 $O$ 点的两侧。引力常量为 $G$ 。 求星球 $\mathrm{B}$ 的线速度大小 $v$; [图1] 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

EX

Astronomy

ZH

multi-modal

Astronomy_270

如图 (i) 所示, 真空中两正点电荷 $\mathrm{A} 、 \mathrm{~B}$ 固定在 $x$ 轴上, 其中 $\mathrm{A}$ 位于坐标原点。 一质量为 $m$ 、电量为 $q$ (电量远小于 $\mathrm{A} 、 \mathrm{~B}$ ) 的带正电小球 $\mathrm{a}$ 仅在电场力作用下, 以大小为 $v_{0}$ 的初速度从 $x=x_{1}$ 处沿 $x$ 轴正方向运动。取无穷远处势能为零, $\mathrm{a}$ 在 $\mathrm{A} 、 \mathrm{~B}$ 间由于受 $\mathrm{A} 、 \mathrm{~B}$ 的电场力作用而具有的电势能 $E_{p}$ 随位置 $x$ 变化关系如图 (ii) 所示, 图中 $E_{N_{1}}$, $E_{2}$ 均为已知, 且 $\mathrm{a}$ 在 $x=x_{2}$ 处受到的电场力为零。 比较 $\mathrm{A} 、 \mathrm{~B}$ 两电荷电量 $Q_{A} 、 Q_{B}$ 的大小关系; [图1] 图(i) [图2] 图(ii) [图3] 图(iii) [图4] 图(iv)

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 如图 (i) 所示, 真空中两正点电荷 $\mathrm{A} 、 \mathrm{~B}$ 固定在 $x$ 轴上, 其中 $\mathrm{A}$ 位于坐标原点。 一质量为 $m$ 、电量为 $q$ (电量远小于 $\mathrm{A} 、 \mathrm{~B}$ ) 的带正电小球 $\mathrm{a}$ 仅在电场力作用下, 以大小为 $v_{0}$ 的初速度从 $x=x_{1}$ 处沿 $x$ 轴正方向运动。取无穷远处势能为零, $\mathrm{a}$ 在 $\mathrm{A} 、 \mathrm{~B}$ 间由于受 $\mathrm{A} 、 \mathrm{~B}$ 的电场力作用而具有的电势能 $E_{p}$ 随位置 $x$ 变化关系如图 (ii) 所示, 图中 $E_{N_{1}}$, $E_{2}$ 均为已知, 且 $\mathrm{a}$ 在 $x=x_{2}$ 处受到的电场力为零。 比较 $\mathrm{A} 、 \mathrm{~B}$ 两电荷电量 $Q_{A} 、 Q_{B}$ 的大小关系; [图1] 图(i) [图2] 图(ii) [图3] 图(iii) [图4] 图(iv) 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

EX

Astronomy

ZH

multi-modal

Astronomy_694

北京时间 2019 年 4 月 10 日, 人类首次利用虚拟射电望远镜, 在紧邻巨椭圆星系 M87 的中心成功捕获世界首张黑洞图像。科学研究表明, 当天体的逃逸速度（即第二宇宙速度, 为第一宇宙速度的 $\sqrt{2}$ 倍）超过光速时, 该天体就是黑洞。已知某天体质量为 $M$,万有引力常量为 $G$, 光速为 $c$, 则要使该天体成为黑洞, 其半径应小于 ( ) A: $\frac{2 G M}{c^{2}}$ B: $\frac{2 c^{2}}{G M}$ C: $\frac{\sqrt{2} G M}{c^{2}}$ D: $\frac{G M}{c^{2}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 北京时间 2019 年 4 月 10 日, 人类首次利用虚拟射电望远镜, 在紧邻巨椭圆星系 M87 的中心成功捕获世界首张黑洞图像。科学研究表明, 当天体的逃逸速度（即第二宇宙速度, 为第一宇宙速度的 $\sqrt{2}$ 倍）超过光速时, 该天体就是黑洞。已知某天体质量为 $M$,万有引力常量为 $G$, 光速为 $c$, 则要使该天体成为黑洞, 其半径应小于 ( ) A: $\frac{2 G M}{c^{2}}$ B: $\frac{2 c^{2}}{G M}$ C: $\frac{\sqrt{2} G M}{c^{2}}$ D: $\frac{G M}{c^{2}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

text-only

Astronomy_1042

The surface of the Sun has a temperature of $\sim 5700 \mathrm{~K}$ yet the solar corona (a very faint region of plasma normally only visible from Earth during a solar eclipse) is considerably hotter at around $10^{6} \mathrm{~K}$. The source of coronal heating is a mystery and so understanding how this might happen is one of several key science objectives of the Solar Orbiter spacecraft. It is equipped with an array of cameras and will take photos of the Sun from distances closer than ever before (other probes will go closer, but none of those have cameras). [figure1] Figure 7: Left: The Sun's corona (coloured green) as viewed in visible light (580-640 nm) taken with the METIS coronagraph instrument onboard Solar Orbiter. The coronagraph is a disc that blocks out the light of the Sun (whose size and position is indicated with the white circle in the middle) so that the faint corona can be seen. This was taken just after first perihelion and is already at a resolution only matched by ground-based telescopes during a solar eclipse - once it gets into the main phase of the mission when it is even closer then its photos will be unrivalled. Credit: METIS Team / ESA \& NASA Right: A high-resolution image from the Extreme Ultraviolet Imager (EUI), taken with the $\mathrm{HRI}_{\mathrm{EUV}}$ telescope just before first perihelion. The circle in the lower right corner indicates the size of Earth for scale. The arrow points to one of the ubiquitous features of the solar surface, called 'campfires', that were discovered by this spacecraft and may play an important role in heating the corona. Credit: EUI Team / ESA \& NASA. Launched in February 2020 (and taken to be at aphelion at launch), it arrived at its first perihelion on $15^{\text {th }}$ June 2020 and has sent back some of the highest resolution images of the surface of the Sun (i.e. the base of the corona) we have ever seen. In them we have identified phenomena nicknamed as 'campfires' (see Fig 7) which are already being considered as a potential major contributor to the mechanism of coronal heating. Later on in its mission it will go in even closer, and so will take photos of the Sun in unprecedented detail. The highest resolution photos are taken with the Extreme Ultraviolet Imager (EUI), which consists of three separate cameras. One of them, the Extreme Ultraviolet High Resolution Imager (HRI $\mathrm{HUV}$ ), is designed to pick up an emission line from highly ionised atoms of iron in the corona. The iron being detected has lost 9 electrons (i.e. $\mathrm{Fe}^{9+}$ ) though is called $\mathrm{Fe} \mathrm{X} \mathrm{('ten')} \mathrm{by} \mathrm{astronomers} \mathrm{(as} \mathrm{Fe} \mathrm{I} \mathrm{is} \mathrm{the} \mathrm{neutral}$ atom). Its presence can be used to work out the temperature of the part of the corona being investigated by the instrument. The photons detected by $\mathrm{HRI}_{\mathrm{EUV}}$ are emitted by a rearrangement of the electrons in the $\mathrm{Fe} \mathrm{X}$ ion, corresponding to a photon energy of $71.0372 \mathrm{eV}$ (where $1 \mathrm{eV}=1.60 \times 10^{-19} \mathrm{~J}$ ). The HRI $\mathrm{HUV}_{\mathrm{EUV}}$ telescope has a $1000^{\prime \prime}$ by $1000^{\prime \prime}$ field of view (FOV, where $1^{\circ}=3600^{\prime \prime}=3600$ arcseconds), an entrance pupil diameter of $47.4 \mathrm{~mm}$, a couple of mirrors that give an effective focal length of $4187 \mathrm{~mm}$, and the image is captured by a CCD with 2048 by 2048 pixels, each of which is 10 by $10 \mu \mathrm{m}$. Although we are viewing the emissions of $\mathrm{Fe} \mathrm{X}$ ions, the vast majority of the plasma in the corona is hydrogen and helium, and the bulk motions of this determine the timescales over which visible phenomena change. In particular, the speed of sound is very important if we do not want motion blur to affect our high resolution images, as this sets the limit on exposure times.a. When it reached first perihelion, radio signals from the probe took $446.58 \mathrm{~s}$ to reach Earth. i. Show that the spacecraft's perihelion is $\approx 0.5$ au, giving your answer to 4 s.f., and hence estimate the launch date, assuming the Earth's orbit is circular. Note that 2020 is a leap year and take 1 year $=365.25$ days. [Hint: You may wish to use a numerical method.]

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is an expression. Here is some context information for this question, which might assist you in solving it: The surface of the Sun has a temperature of $\sim 5700 \mathrm{~K}$ yet the solar corona (a very faint region of plasma normally only visible from Earth during a solar eclipse) is considerably hotter at around $10^{6} \mathrm{~K}$. The source of coronal heating is a mystery and so understanding how this might happen is one of several key science objectives of the Solar Orbiter spacecraft. It is equipped with an array of cameras and will take photos of the Sun from distances closer than ever before (other probes will go closer, but none of those have cameras). [figure1] Figure 7: Left: The Sun's corona (coloured green) as viewed in visible light (580-640 nm) taken with the METIS coronagraph instrument onboard Solar Orbiter. The coronagraph is a disc that blocks out the light of the Sun (whose size and position is indicated with the white circle in the middle) so that the faint corona can be seen. This was taken just after first perihelion and is already at a resolution only matched by ground-based telescopes during a solar eclipse - once it gets into the main phase of the mission when it is even closer then its photos will be unrivalled. Credit: METIS Team / ESA \& NASA Right: A high-resolution image from the Extreme Ultraviolet Imager (EUI), taken with the $\mathrm{HRI}_{\mathrm{EUV}}$ telescope just before first perihelion. The circle in the lower right corner indicates the size of Earth for scale. The arrow points to one of the ubiquitous features of the solar surface, called 'campfires', that were discovered by this spacecraft and may play an important role in heating the corona. Credit: EUI Team / ESA \& NASA. Launched in February 2020 (and taken to be at aphelion at launch), it arrived at its first perihelion on $15^{\text {th }}$ June 2020 and has sent back some of the highest resolution images of the surface of the Sun (i.e. the base of the corona) we have ever seen. In them we have identified phenomena nicknamed as 'campfires' (see Fig 7) which are already being considered as a potential major contributor to the mechanism of coronal heating. Later on in its mission it will go in even closer, and so will take photos of the Sun in unprecedented detail. The highest resolution photos are taken with the Extreme Ultraviolet Imager (EUI), which consists of three separate cameras. One of them, the Extreme Ultraviolet High Resolution Imager (HRI $\mathrm{HUV}$ ), is designed to pick up an emission line from highly ionised atoms of iron in the corona. The iron being detected has lost 9 electrons (i.e. $\mathrm{Fe}^{9+}$ ) though is called $\mathrm{Fe} \mathrm{X} \mathrm{('ten')} \mathrm{by} \mathrm{astronomers} \mathrm{(as} \mathrm{Fe} \mathrm{I} \mathrm{is} \mathrm{the} \mathrm{neutral}$ atom). Its presence can be used to work out the temperature of the part of the corona being investigated by the instrument. The photons detected by $\mathrm{HRI}_{\mathrm{EUV}}$ are emitted by a rearrangement of the electrons in the $\mathrm{Fe} \mathrm{X}$ ion, corresponding to a photon energy of $71.0372 \mathrm{eV}$ (where $1 \mathrm{eV}=1.60 \times 10^{-19} \mathrm{~J}$ ). The HRI $\mathrm{HUV}_{\mathrm{EUV}}$ telescope has a $1000^{\prime \prime}$ by $1000^{\prime \prime}$ field of view (FOV, where $1^{\circ}=3600^{\prime \prime}=3600$ arcseconds), an entrance pupil diameter of $47.4 \mathrm{~mm}$, a couple of mirrors that give an effective focal length of $4187 \mathrm{~mm}$, and the image is captured by a CCD with 2048 by 2048 pixels, each of which is 10 by $10 \mu \mathrm{m}$. Although we are viewing the emissions of $\mathrm{Fe} \mathrm{X}$ ions, the vast majority of the plasma in the corona is hydrogen and helium, and the bulk motions of this determine the timescales over which visible phenomena change. In particular, the speed of sound is very important if we do not want motion blur to affect our high resolution images, as this sets the limit on exposure times. problem: a. When it reached first perihelion, radio signals from the probe took $446.58 \mathrm{~s}$ to reach Earth. i. Show that the spacecraft's perihelion is $\approx 0.5$ au, giving your answer to 4 s.f., and hence estimate the launch date, assuming the Earth's orbit is circular. Note that 2020 is a leap year and take 1 year $=365.25$ days. [Hint: You may wish to use a numerical method.] All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is an expression without equals signs, e.g. ANSWER=\frac{1}{2} g t^2

EX

Astronomy

EN

multi-modal

Astronomy_6

2022 年 11 月 29 日 23 时 08 分, 搭载着神舟十五号载人飞船的长征二号 $F$ 遥十五运载火箭在酒泉卫星发射中心升空, 11 月 30 日 5 时 42 分, 神舟十五号载人飞船与天和核心舱成功完成自主交会对接。如图为神舟十五号的发射与交会对接过程示意图，图中 I 为近地圆轨道, 其轨道半径可认为等于地球半径 $R$, II 为椭圆变轨轨道, III 为天和核心舱所在轨道, 其轨道半径为 $r_{0}, P 、 Q$ 分别为轨道 II 与 I、III 轨道的交会点, 已知神舟十五号的质量为 $m_{0}$, 地球表面重力加速度为 $g$, 引力常量为 $G$, 若取两物体相距无 穷远时的引力势能为零, 一个质量为 $m$ 的质点距质量为 $M$ 的引力中心为 $r$ 时, 其万有引力势能表达式为 $E_{\mathrm{P}}=-\frac{G M m}{r}$ (式中 $G$ 为引力常量)。求: 要使神舟十五号从轨道 I 迁移到轨道 III, 所要提供的最小能量 [图1]

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 2022 年 11 月 29 日 23 时 08 分, 搭载着神舟十五号载人飞船的长征二号 $F$ 遥十五运载火箭在酒泉卫星发射中心升空, 11 月 30 日 5 时 42 分, 神舟十五号载人飞船与天和核心舱成功完成自主交会对接。如图为神舟十五号的发射与交会对接过程示意图，图中 I 为近地圆轨道, 其轨道半径可认为等于地球半径 $R$, II 为椭圆变轨轨道, III 为天和核心舱所在轨道, 其轨道半径为 $r_{0}, P 、 Q$ 分别为轨道 II 与 I、III 轨道的交会点, 已知神舟十五号的质量为 $m_{0}$, 地球表面重力加速度为 $g$, 引力常量为 $G$, 若取两物体相距无 穷远时的引力势能为零, 一个质量为 $m$ 的质点距质量为 $M$ 的引力中心为 $r$ 时, 其万有引力势能表达式为 $E_{\mathrm{P}}=-\frac{G M m}{r}$ (式中 $G$ 为引力常量)。求: 要使神舟十五号从轨道 I 迁移到轨道 III, 所要提供的最小能量 [图1] 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

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Astronomy_780

What is a synodic month? A: The time between two full moons. B: The time between two Venus appearances. C: The time for $1 / 12$ th Earth orbit around the Sun. D: The time for one Earth rotation in respect to the Sun.

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: What is a synodic month? A: The time between two full moons. B: The time between two Venus appearances. C: The time for $1 / 12$ th Earth orbit around the Sun. D: The time for one Earth rotation in respect to the Sun. You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

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Astronomy_351

天问一号火星探测器的发射标志着我国的航天事业迈进了新时代, 设地球绕太阳的公转周期为 $T$, 环绕太阳公转的轨道半径为 $r_{1}$, 火星环绕太阳公转的轨道半径为 $r_{2}$, 火星的半径为 $R$, 万有引力常量为 $G$, 下列说法正确的是 ( ) A: 太阳的质量为 $\frac{4 \pi^{2} r_{2}^{3}}{G T^{2}}$ B: 火星绕太阳公转的角速度大小为 $\frac{2 \pi}{T}\left(\frac{r_{2}}{r_{1}}\right)^{\frac{3}{2}}$ C: 火星表面的重力加速度大小为 $\frac{4 \pi^{2} r_{1}^{3}}{R^{2} T^{2}}$ D: 从火星与地球相距最远到地球与火星相距最近的最短时间为 $\frac{r_{2}^{\frac{3}{2}} T}{2\left(r_{2}^{\frac{3}{2}}-r_{1}^{\frac{3}{2}}\right)}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 天问一号火星探测器的发射标志着我国的航天事业迈进了新时代, 设地球绕太阳的公转周期为 $T$, 环绕太阳公转的轨道半径为 $r_{1}$, 火星环绕太阳公转的轨道半径为 $r_{2}$, 火星的半径为 $R$, 万有引力常量为 $G$, 下列说法正确的是 ( ) A: 太阳的质量为 $\frac{4 \pi^{2} r_{2}^{3}}{G T^{2}}$ B: 火星绕太阳公转的角速度大小为 $\frac{2 \pi}{T}\left(\frac{r_{2}}{r_{1}}\right)^{\frac{3}{2}}$ C: 火星表面的重力加速度大小为 $\frac{4 \pi^{2} r_{1}^{3}}{R^{2} T^{2}}$ D: 从火星与地球相距最远到地球与火星相距最近的最短时间为 $\frac{r_{2}^{\frac{3}{2}} T}{2\left(r_{2}^{\frac{3}{2}}-r_{1}^{\frac{3}{2}}\right)}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy_430

如图所示, 甲、乙两卫星在某行星的球心的同一平面内做圆周运动, 某时刻恰好处于行星上 $A$ 点的正上方, 从该时刻算起, 在同一段时间内, 甲卫星恰好又有 5 次经过 $A$点的正上方, 乙卫星恰好又有 3 次经过 $A$ 点的正上方, 不计行星自转的影响, 下列关于这两颗卫星的说法正确的是 [图1] A: 甲、乙两卫量的周期之比为 $2: 3$ B: 甲、乙两卫星的角速度之比为 $3: 5$ C: 甲、乙两卫星的轨道半径之比为 $\sqrt[3]{\frac{9}{25}}$ D: 若甲、乙两卫星质量相同, 则甲的机械能大于乙的机械能

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 如图所示, 甲、乙两卫星在某行星的球心的同一平面内做圆周运动, 某时刻恰好处于行星上 $A$ 点的正上方, 从该时刻算起, 在同一段时间内, 甲卫星恰好又有 5 次经过 $A$点的正上方, 乙卫星恰好又有 3 次经过 $A$ 点的正上方, 不计行星自转的影响, 下列关于这两颗卫星的说法正确的是 [图1] A: 甲、乙两卫量的周期之比为 $2: 3$ B: 甲、乙两卫星的角速度之比为 $3: 5$ C: 甲、乙两卫星的轨道半径之比为 $\sqrt[3]{\frac{9}{25}}$ D: 若甲、乙两卫星质量相同, 则甲的机械能大于乙的机械能 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy_567

地球质量大约是月球质量的 81 倍, 地月距离约为 38 万千米, 两者中心连线上有一个被称作“拉格朗日点” 的位置, 一飞行器处于该点, 在几乎不消耗燃料的情况下与月球同步绕地球做圆周运动, 则这个点到地球的距离约为 ( ) A: 3.8 万千米 B: 5.8 万千米 C: 32 万千米 D: 34 万千米

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 地球质量大约是月球质量的 81 倍, 地月距离约为 38 万千米, 两者中心连线上有一个被称作“拉格朗日点” 的位置, 一飞行器处于该点, 在几乎不消耗燃料的情况下与月球同步绕地球做圆周运动, 则这个点到地球的距离约为 ( ) A: 3.8 万千米 B: 5.8 万千米 C: 32 万千米 D: 34 万千米 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy_525

2018 年 3 月 30 日, 我国在西昌卫星发射中心用长征三号乙运载火箭, 以“一箭双星”方式成功发射第三十、三十一颗北斗导航卫星. 已知“北斗第三十颗导航卫星”做匀速圆周运动的轨道半径小于地球同步卫星轨道半径, 运行速度为 $v$, 向心加速度为 $a$;地球表面的重力加速度为 $g$, 引力常量为 $G$. 下列判断正确的是 ( ) A: 该导航卫星的运行周期大于 24 小时 B: 地球质量为 $\frac{v^{4}}{G a}$ C: 该导航卫星的轨道半径与地球半径之比为 $\sqrt{g}: \sqrt{a}$ D: 该导航卫星的运行速度与地球第一宇宙速度之比 $\sqrt{a}: \sqrt{g}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 2018 年 3 月 30 日, 我国在西昌卫星发射中心用长征三号乙运载火箭, 以“一箭双星”方式成功发射第三十、三十一颗北斗导航卫星. 已知“北斗第三十颗导航卫星”做匀速圆周运动的轨道半径小于地球同步卫星轨道半径, 运行速度为 $v$, 向心加速度为 $a$;地球表面的重力加速度为 $g$, 引力常量为 $G$. 下列判断正确的是 ( ) A: 该导航卫星的运行周期大于 24 小时 B: 地球质量为 $\frac{v^{4}}{G a}$ C: 该导航卫星的轨道半径与地球半径之比为 $\sqrt{g}: \sqrt{a}$ D: 该导航卫星的运行速度与地球第一宇宙速度之比 $\sqrt{a}: \sqrt{g}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

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Astronomy_138

我国的“天链一号”地球同步轨道卫星, 可为载人航天器及中、低轨道卫星提供数据通讯服务。如图为“天链一号” $\mathrm{a}$ 、赤道平面内的低轨道卫星 b、地球三者的位置关系示意图, $O$ 为地心, 地球相对卫星 $\mathrm{a} 、 \mathrm{~b}$ 的张角分别为 $\theta_{1}$ 和 $\theta_{2}$ ( $\theta_{2}$ 图中未标出), 卫星 $\mathrm{a}$ 的轨道半径是 $\mathrm{b}$ 的 4 倍, 已知卫星 $\mathrm{a} 、 \mathrm{~b}$ 绕地球同向运行, 卫星 $\mathrm{a}$ 周期为 $T$, 在运行过程中由于地球的遮挡, 卫星 $\mathrm{b}$ 会进入与卫星 $\mathrm{a}$ 通讯的盲区。卫星间的通讯信号视为沿直线传播，忽略信号传输时间，下列分析正确的是（ ） [图1] A: 张角 $\theta_{1}$ 和 $\theta_{2}$ 满足 $\sin \theta_{2}=4 \sin \theta_{1}$ B: 卫星 $\mathrm{b}$ 的周期为 $\frac{T}{8}$ C: 卫星 $\mathrm{b}$ 每次在盲区运行的时间 $\frac{\theta_{1}+\theta_{2}}{14 \pi} T$ D: 卫星 $\mathrm{b}$ 每次在盲区运行的时间为 $\frac{\theta_{1}+\theta_{2}}{16 \pi} T$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 我国的“天链一号”地球同步轨道卫星, 可为载人航天器及中、低轨道卫星提供数据通讯服务。如图为“天链一号” $\mathrm{a}$ 、赤道平面内的低轨道卫星 b、地球三者的位置关系示意图, $O$ 为地心, 地球相对卫星 $\mathrm{a} 、 \mathrm{~b}$ 的张角分别为 $\theta_{1}$ 和 $\theta_{2}$ ( $\theta_{2}$ 图中未标出), 卫星 $\mathrm{a}$ 的轨道半径是 $\mathrm{b}$ 的 4 倍, 已知卫星 $\mathrm{a} 、 \mathrm{~b}$ 绕地球同向运行, 卫星 $\mathrm{a}$ 周期为 $T$, 在运行过程中由于地球的遮挡, 卫星 $\mathrm{b}$ 会进入与卫星 $\mathrm{a}$ 通讯的盲区。卫星间的通讯信号视为沿直线传播，忽略信号传输时间，下列分析正确的是（ ） [图1] A: 张角 $\theta_{1}$ 和 $\theta_{2}$ 满足 $\sin \theta_{2}=4 \sin \theta_{1}$ B: 卫星 $\mathrm{b}$ 的周期为 $\frac{T}{8}$ C: 卫星 $\mathrm{b}$ 每次在盲区运行的时间 $\frac{\theta_{1}+\theta_{2}}{14 \pi} T$ D: 卫星 $\mathrm{b}$ 每次在盲区运行的时间为 $\frac{\theta_{1}+\theta_{2}}{16 \pi} T$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

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Astronomy

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multi-modal

Astronomy_599

宇宙中存在一些离其他恒星较远的、由质量相等的三颗星组成的三星系统, 可忽略其他星体对三星系统的影响。稳定的三星系统存在两种基本形式: 一种是三颗星位于同一直线上, 两颗星围绕中央星在同一半径为 $R$ 的轨道上运行, 如图甲所示, 周期为 $T_{1}$;另一种是三颗星位于边长为 $R$ 的等边三角形的三个顶点上, 并沿等边三角形的外接圆运行, 如图乙所示, 周期为 $T_{2}$ 。则 $T_{1}: T_{2}$ 为 ( ) [图1] 甲 [图2] 乙 A: $\sqrt{\frac{3}{5}}$ B: $2 \sqrt{\frac{3}{5}}$ C: $3 \sqrt{\frac{3}{5}}$ D: $4 \sqrt{\frac{3}{5}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 宇宙中存在一些离其他恒星较远的、由质量相等的三颗星组成的三星系统, 可忽略其他星体对三星系统的影响。稳定的三星系统存在两种基本形式: 一种是三颗星位于同一直线上, 两颗星围绕中央星在同一半径为 $R$ 的轨道上运行, 如图甲所示, 周期为 $T_{1}$;另一种是三颗星位于边长为 $R$ 的等边三角形的三个顶点上, 并沿等边三角形的外接圆运行, 如图乙所示, 周期为 $T_{2}$ 。则 $T_{1}: T_{2}$ 为 ( ) [图1] 甲 [图2] 乙 A: $\sqrt{\frac{3}{5}}$ B: $2 \sqrt{\frac{3}{5}}$ C: $3 \sqrt{\frac{3}{5}}$ D: $4 \sqrt{\frac{3}{5}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy_747

An object's spectral energy distribution (SED) is formally given by ... A: $d E / d \lambda$ B: $d E / d t$ C: $d z / d \lambda$ D: $d z / d t$

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: An object's spectral energy distribution (SED) is formally given by ... A: $d E / d \lambda$ B: $d E / d t$ C: $d z / d \lambda$ D: $d z / d t$ You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

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Astronomy_414

由三颗星体构成的系统, 忽略其他星体对它们的作用, 存在着一种运动形式: 三颗星体在相互之间的万有引力作用下, 分别位于等边三角形的三个顶点上, 绕某一共同的圆心 $\mathrm{O}$ 在三角形所在的平面内做相同角速度的圆周运动 (图示为 $\mathrm{A} 、 \mathrm{~B} 、 \mathrm{C}$ 三颗星体质量不相同时的一般情况). 若 $\mathrm{A}$ 星体质量为 $2 \mathrm{~m}, \mathrm{~B} 、 \mathrm{C}$ 两星体的质量均为 $\mathrm{m}$, 三角形边长为 $\mathrm{a}$. 则 ( ) [图1] A: B 星体所受的合力与 $\mathrm{A}$ 星体所受合力之比为 $1: 2$ B: 圆心 $\mathrm{O}$ 与 $\mathrm{B}$ 的连线与 $\mathrm{BC}$ 夹角 $\theta$ 的正切值为 $\frac{\sqrt{3}}{2}$ C: A、B 星体做圆周运动的线速度大小之比为 $\sqrt{3}: \sqrt{7}$ D: 此三星体做圆周运动的角速度大小为 $2 \sqrt{\frac{G m^{2}}{a^{3}}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 由三颗星体构成的系统, 忽略其他星体对它们的作用, 存在着一种运动形式: 三颗星体在相互之间的万有引力作用下, 分别位于等边三角形的三个顶点上, 绕某一共同的圆心 $\mathrm{O}$ 在三角形所在的平面内做相同角速度的圆周运动 (图示为 $\mathrm{A} 、 \mathrm{~B} 、 \mathrm{C}$ 三颗星体质量不相同时的一般情况). 若 $\mathrm{A}$ 星体质量为 $2 \mathrm{~m}, \mathrm{~B} 、 \mathrm{C}$ 两星体的质量均为 $\mathrm{m}$, 三角形边长为 $\mathrm{a}$. 则 ( ) [图1] A: B 星体所受的合力与 $\mathrm{A}$ 星体所受合力之比为 $1: 2$ B: 圆心 $\mathrm{O}$ 与 $\mathrm{B}$ 的连线与 $\mathrm{BC}$ 夹角 $\theta$ 的正切值为 $\frac{\sqrt{3}}{2}$ C: A、B 星体做圆周运动的线速度大小之比为 $\sqrt{3}: \sqrt{7}$ D: 此三星体做圆周运动的角速度大小为 $2 \sqrt{\frac{G m^{2}}{a^{3}}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

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Astronomy

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multi-modal

Astronomy_92

“天问一号”探测器需要通过霍曼转移轨道从地球发送到火星, 地球轨道和火星轨道看成圆形轨道, 此时霍曼转移轨道是一个近日点 $M$ 和远日点 $P$ 都与地球轨道、火星轨道相切的椭圆轨道 (如图所示)。在近日点短暂点火后“天问一号”进入霍曼转移轨道，接着“天问一号”沿着这个轨道直至抵达远日点, 然后再次点火进入火星轨道。已知万有引力常量为 $G$, 太阳质量为 $m$, 地球轨道和火星轨道半径分别为 $r$ 和 $R$, 地球、火星、 “天问一号”运行方向都为逆时针方向。下列说法正确的是（） [图1] A: 两次点火时喷气方向都与运动方向相同 B: 两次点火之间的时间为 $\frac{\pi}{2 \sqrt{2}} \sqrt{\frac{(r+R)^{3}}{G m}}$ C: “天问一号”与太阳连线单位时间在地球轨道上扫过的面积等于在火星轨道上扫过的面积 D: “天问一号”在转移轨道上近日点的速度大小 $v_{1}$ 比远日点的速度大小 $v_{2}$ 大, 且满足 $\frac{v_{1}}{v_{2}}=\sqrt{\frac{R}{r}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: “天问一号”探测器需要通过霍曼转移轨道从地球发送到火星, 地球轨道和火星轨道看成圆形轨道, 此时霍曼转移轨道是一个近日点 $M$ 和远日点 $P$ 都与地球轨道、火星轨道相切的椭圆轨道 (如图所示)。在近日点短暂点火后“天问一号”进入霍曼转移轨道，接着“天问一号”沿着这个轨道直至抵达远日点, 然后再次点火进入火星轨道。已知万有引力常量为 $G$, 太阳质量为 $m$, 地球轨道和火星轨道半径分别为 $r$ 和 $R$, 地球、火星、 “天问一号”运行方向都为逆时针方向。下列说法正确的是（） [图1] A: 两次点火时喷气方向都与运动方向相同 B: 两次点火之间的时间为 $\frac{\pi}{2 \sqrt{2}} \sqrt{\frac{(r+R)^{3}}{G m}}$ C: “天问一号”与太阳连线单位时间在地球轨道上扫过的面积等于在火星轨道上扫过的面积 D: “天问一号”在转移轨道上近日点的速度大小 $v_{1}$ 比远日点的速度大小 $v_{2}$ 大, 且满足 $\frac{v_{1}}{v_{2}}=\sqrt{\frac{R}{r}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

multi-modal

Astronomy_936

In the UK we use the Gregorian calendar; it is a solar calendar so that a year corresponds to the time to orbit the Sun once, where 1 solar year is $\approx 365.25$ days. Several cultures use a lunar calendar, where each month is determined by the time it takes to go from New Moon to New Moon, and have a lunar year that is exactly 12 lunar months. An example of this is the Islamic calendar. Since the length of a lunar month (29.53 days) is a little shorter than the average month length in our solar calendar (see Figure 3), it means the start date of each month in the Islamic calendar is not tied to the seasons and gradually moves earlier in the solar year. [figure1] Figure 3: All the moon phases in May 2022, showing that the time measured from New Moon to New Moon (a lunar month) is shorter than a month in the Gregorian calendar. Credit: MoonConnection.com In some Islamic countries, odd-numbered months have 30 days and even-numbered months have 29 days. How often will leap years be in this system (where the twelfth month is also 30 days)? [This is analogous to the Gregorian system where there is a leap year roughly once every 4 years.]

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. problem: In the UK we use the Gregorian calendar; it is a solar calendar so that a year corresponds to the time to orbit the Sun once, where 1 solar year is $\approx 365.25$ days. Several cultures use a lunar calendar, where each month is determined by the time it takes to go from New Moon to New Moon, and have a lunar year that is exactly 12 lunar months. An example of this is the Islamic calendar. Since the length of a lunar month (29.53 days) is a little shorter than the average month length in our solar calendar (see Figure 3), it means the start date of each month in the Islamic calendar is not tied to the seasons and gradually moves earlier in the solar year. [figure1] Figure 3: All the moon phases in May 2022, showing that the time measured from New Moon to New Moon (a lunar month) is shorter than a month in the Gregorian calendar. Credit: MoonConnection.com In some Islamic countries, odd-numbered months have 30 days and even-numbered months have 29 days. How often will leap years be in this system (where the twelfth month is also 30 days)? [This is analogous to the Gregorian system where there is a leap year roughly once every 4 years.] All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of lunar years, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

multi-modal

Astronomy_37

2020 年 7 月 23 日, 我国火星探测器 “天问一号” 发射成功, 2021 年 1 月 28 日, “天问一号” 飞行里程突破四亿千米, 图甲是火星探测器的运行路线图。假设探测器经过多次变轨后登陆火星的轨迹变化可抽象为如图乙所示, 探测器先在轨道 I 上运动, 经过 $P$点启动变轨发动机切换到圆轨道 II 上运动, 经过一段时间后, 再次经过 $P$ 点时启动变轨发动机切换到椭圆轨道III上运动。轨道上的 $P 、 Q 、 \mathrm{~S}$ 三点与火星中心位于同一直线上, $P 、 Q$ 两点分别是粗圆轨道的远火星点和近火星点, 且 $P Q=2 Q S=2 l$ 。除了变轨瞬间,探测器在轨道上运行时均处于无动力航行状态。探测器在轨道 I、II、III 上经过 $P$ 点的速度分别为 $v_{1} 、 v_{2} 、 v_{3}$, 下列说法正确的是 ( ) [图1] 甲 [图2] 乙 A: $v_{1}<v_{2}<v_{3}$ B: 探测器在轨道 III 上从 $P$ 点运动到 $Q$ 点的过程中速率变大 C: 探测器在轨道 III 上运动时, 经过 $P$ 点的加速度为 $\frac{2 v_{2}^{2}}{3 l}$ D: 探测器在轨道 II 上由 $P$ 点运动到 $\mathrm{S}$ 点与探测器在轨道III上由 $P$ 点运动到 $Q$ 点的时间之比为 9: 4

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 2020 年 7 月 23 日, 我国火星探测器 “天问一号” 发射成功, 2021 年 1 月 28 日, “天问一号” 飞行里程突破四亿千米, 图甲是火星探测器的运行路线图。假设探测器经过多次变轨后登陆火星的轨迹变化可抽象为如图乙所示, 探测器先在轨道 I 上运动, 经过 $P$点启动变轨发动机切换到圆轨道 II 上运动, 经过一段时间后, 再次经过 $P$ 点时启动变轨发动机切换到椭圆轨道III上运动。轨道上的 $P 、 Q 、 \mathrm{~S}$ 三点与火星中心位于同一直线上, $P 、 Q$ 两点分别是粗圆轨道的远火星点和近火星点, 且 $P Q=2 Q S=2 l$ 。除了变轨瞬间,探测器在轨道上运行时均处于无动力航行状态。探测器在轨道 I、II、III 上经过 $P$ 点的速度分别为 $v_{1} 、 v_{2} 、 v_{3}$, 下列说法正确的是 ( ) [图1] 甲 [图2] 乙 A: $v_{1}<v_{2}<v_{3}$ B: 探测器在轨道 III 上从 $P$ 点运动到 $Q$ 点的过程中速率变大 C: 探测器在轨道 III 上运动时, 经过 $P$ 点的加速度为 $\frac{2 v_{2}^{2}}{3 l}$ D: 探测器在轨道 II 上由 $P$ 点运动到 $\mathrm{S}$ 点与探测器在轨道III上由 $P$ 点运动到 $Q$ 点的时间之比为 9: 4 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_841

VOIDED Austin is now observing LARES, but in this problem he is allowed to observe LARES at any declination. Assume that LARES is a uniform spherical ball, and for the purposes of this problem, assume that LARES is a perfect blackbody; i.e. THA- $18 \mathrm{~N}$ has an albedo of 0 . What is the brightest apparent magnitude Austin can observe LARES? You can use the fact that the Sun's apparent magnitude is -26.74 . A: 6.78 B: 7.77 C: 8.76 D: 9.75 E: 10.74

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: VOIDED Austin is now observing LARES, but in this problem he is allowed to observe LARES at any declination. Assume that LARES is a uniform spherical ball, and for the purposes of this problem, assume that LARES is a perfect blackbody; i.e. THA- $18 \mathrm{~N}$ has an albedo of 0 . What is the brightest apparent magnitude Austin can observe LARES? You can use the fact that the Sun's apparent magnitude is -26.74 . A: 6.78 B: 7.77 C: 8.76 D: 9.75 E: 10.74 You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D, E].

SC

Astronomy

EN

text-only

Astronomy_233

假设地球是一半径为 $R$ 、质量分布均匀的球体，设想以地心为圆心，在半径为 $r$ 处开丵一圆形隧道, 在隧道内有一小球绕地心做匀速圆周运动, 且对隧道内外壁的压力为零, 如图所示。已知质量分布均匀的球壳对壳内物体的引力为零。地球的第一宇宙速度为 $v_{1}$ ，小球的线速度为 $v_{2}$, 则 $\frac{v_{1}}{v_{2}}$ 等于 ( ) [图1] A: $\frac{r}{R}$ B: $\frac{R}{r}$ C: $\left(\frac{r}{R}\right)^{2}$ D: $\left(\frac{R}{r}\right)^{2}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 假设地球是一半径为 $R$ 、质量分布均匀的球体，设想以地心为圆心，在半径为 $r$ 处开丵一圆形隧道, 在隧道内有一小球绕地心做匀速圆周运动, 且对隧道内外壁的压力为零, 如图所示。已知质量分布均匀的球壳对壳内物体的引力为零。地球的第一宇宙速度为 $v_{1}$ ，小球的线速度为 $v_{2}$, 则 $\frac{v_{1}}{v_{2}}$ 等于 ( ) [图1] A: $\frac{r}{R}$ B: $\frac{R}{r}$ C: $\left(\frac{r}{R}\right)^{2}$ D: $\left(\frac{R}{r}\right)^{2}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

multi-modal

Astronomy_683

某天体科学家在太阳系外发现一颗类地球行星, 这颗类地行星绕中心恒星做圆周运动, 公转的周期为 146 天, 体积是地球体积的 8 倍, 行星表面的重力加速度是地球表面重力加速度的 2 倍, 它与中心恒星间的距离跟地球和太阳的距离相近。地球公转周期为 365 天，类地行星和地球均看作密度均匀的球体。求: 这颗类地行星与地球的平均密度之比 $\frac{\rho_{\text {行 }}}{\rho_{\text {地 }}}$ 为多少?

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个数值。 问题: 某天体科学家在太阳系外发现一颗类地球行星, 这颗类地行星绕中心恒星做圆周运动, 公转的周期为 146 天, 体积是地球体积的 8 倍, 行星表面的重力加速度是地球表面重力加速度的 2 倍, 它与中心恒星间的距离跟地球和太阳的距离相近。地球公转周期为 365 天，类地行星和地球均看作密度均匀的球体。求: 这颗类地行星与地球的平均密度之比 $\frac{\rho_{\text {行 }}}{\rho_{\text {地 }}}$ 为多少? 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是数值。

NV

Astronomy

ZH

text-only

Astronomy_1114

In July 1969 the mission Apollo 11 was the first to successfully allow humans to walk on the Moon. This was an incredible achievement as the engineering necessary to make it a possibility was an order of magnitude more complex than anything that had come before. The Apollo 11 spacecraft was launched atop the Saturn V rocket, which still stands as the most powerful rocket ever made. [figure1] Figure 1: Left: The launch of Apollo 11 upon the Saturn V rocket. Credit: NASA. Right: Showing the three stages of the Saturn V rocket (each detached once its fuel was expended), plus the Apollo spacecraft on top (containing three astronauts) which was delivered into a translunar orbit. At the base of the rocket is a person to scale, emphasising the enormous size of the rocket. Credit: Encyclopaedia Britannica. | Stage | Initial Mass $(\mathrm{t})$ | Final mass $(\mathrm{t})$ | $I_{\mathrm{sp}}(\mathrm{s})$ | Burn duration $(\mathrm{s})$ | | :---: | :---: | :---: | :---: | :---: | | S-IC | 2283.9 | 135.6 | 263 | 168 | | S-II | 483.7 | 39.9 | 421 | 384 | | S-IV (Burn 1) | 121.0 | - | 421 | 147 | | S-IV (Burn 2) | - | 13.2 | 421 | 347 | | Apollo Spacecraft | 49.7 | - | - | - | Table 1: Data about each stage of the rocket used to launch the Apollo 11 spacecraft into a translunar orbit. Masses are given in tonnes $(1 \mathrm{t}=1000 \mathrm{~kg}$ ) and for convenience include the interstage parts of the rocket too. The specific impulse, $I_{\mathrm{sp}}$, of the stage is given at sea level atmospheric pressure for S-IC and for a vacuum for S-II and S-IVB. The Saturn V rocket consisted of three stages (see Fig 1), since this was the only practical way to get the Apollo spacecraft up to the speed necessary to make the transfer to the Moon. When fully fueled the mass of the total rocket was immense, and lots of that fuel was necessary to simply lift the fuel of the later stages into high altitude - in total about $3000 \mathrm{t}(1$ tonne, $\mathrm{t}=1000 \mathrm{~kg}$ ) of rocket on the launchpad was required to send about $50 \mathrm{t}$ on a mission to the Moon. The first stage (called S-IC) was the heaviest, the second (called S-II) was considerably lighter, and the third stage (called S-IVB) was fired twice - the first to get the spacecraft into a circular 'parking' orbit around the Earth where various safety checks were made, whilst the second burn was to get the spacecraft on its way to the Moon. Once each rocket stage was fully spent it was detached from the rest of the rocket before the next stage ignited. Data about each stage is given in Table 1. The thrust of the rocket is given as $$ F=-I_{\mathrm{sp}} g_{0} \dot{m} $$ where the specific impulse, $I_{\mathrm{sp}}$, of each stage is a constant related to the type of fuel used and the shape of the rocket nozzle, $g_{0}$ is the gravitational field strength of the Earth at sea level (i.e. $g_{0}=9.81 \mathrm{~m} \mathrm{~s}^{-2}$ ) and $\dot{m} \equiv \mathrm{d} m / \mathrm{d} t$ is the rate of change of mass of the rocket with time. The thrust generated by the first two stages (S-IC and S-II) can be taken to be constant. However, the thrust generated by the third stage (S-IVB) varied in order to give a constant acceleration (taken to be the same throughout both burns of the rocket). By the end of the second burn the Apollo spacecraft, apart from a few short burns to give mid-course corrections, coasted all the way to the far side of the Moon where the engines were then fired again to circularise the orbit. All of the early Apollo missions were on a orbit known as a 'free-return trajectory', meaning that if there was a problem then they were already on an orbit that would take them back to Earth after passing around the Moon. The real shape of such a trajectory (in a rotating frame of reference) is like a stretched figure of 8 and is shown in the top panel of Fig 2. To calculate this precisely is non-trivial and required substantial computing power in the 1960s. However, we can have two simplified models that can be used to estimate the duration of the translunar coast, and they are shown in the bottom panel of the Fig 2. The first is a Hohmann transfer orbit (dashed line), which is a single ellipse with the Earth at one focus. In this model the gravitational effect of the Moon is ignored, so the spacecraft travels from A (the perigee) to B (the apogee). The second (solid line) takes advantage of a 'patched conics' approach by having two ellipses whose apoapsides coincide at point $\mathrm{C}$ where the gravitational force on the spacecraft is equal from both the Earth and the Moon. The first ellipse has a periapsis at A and ignores the gravitational effect of the Moon, whilst the second ellipse has a periapsis at B and ignores the gravitational effect of the Earth. If the spacecraft trajectory and lunar orbit are coplanar and the Moon is in a circular orbit around the Earth then the time to travel from $\mathrm{A}$ to $\mathrm{B}$ via $\mathrm{C}$ is double the value attained if taking into account the gravitational forces of the Earth and Moon together throughout the journey, which is a much better estimate of the time of a real translunar coast. [figure2] Figure 2: Top: The real shape of a translunar free-return trajectory, with the Earth on the left and the Moon on the right (orbiting around the Earth in an anti-clockwise direction). This diagram (and the one below) is shown in a co-ordinate system co-rotating with the Earth and is not to scale. Credit: NASA. Bottom: Two simplified ways of modelling the translunar trajectory. The simplest is a Hohmann transfer orbit (dashed line, outer ellipse), which is an ellipse that has the Earth at one focus and ignores the gravitational effect of the Moon. A better model (solid line, inner ellipses) of the Apollo trajectory is the use of two ellipses that meet at point $\mathrm{C}$ where the gravitational forces of the Earth and Moon on the spacecraft are equal. For the Apollo 11 journey, the end of the second burn of the S-IVB rocket (point A) was $334 \mathrm{~km}$ above the surface of the Earth, and the end of the translunar coast (point B) was $161 \mathrm{~km}$ above the surface of the Moon. The distance between the centres of mass of the Earth and the Moon at the end of the translunar coast was $3.94 \times 10^{8} \mathrm{~m}$. Take the radius of the Earth to be $6370 \mathrm{~km}$, the radius of the Moon to be $1740 \mathrm{~km}$, and the mass of the Moon to be $7.35 \times 10^{22} \mathrm{~kg}$.d. For the patched conics approach (solid lines): ii. Calculate the speed of the spacecraft at point A and point B. Give your answer in $\mathrm{km} \mathrm{s}^{-1}$ and as a percentage of the escape speed of the spacecraft at that distance from the relevant closest gravitational body. Comment on what this implies for the eccentricity of the orbits.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: In July 1969 the mission Apollo 11 was the first to successfully allow humans to walk on the Moon. This was an incredible achievement as the engineering necessary to make it a possibility was an order of magnitude more complex than anything that had come before. The Apollo 11 spacecraft was launched atop the Saturn V rocket, which still stands as the most powerful rocket ever made. [figure1] Figure 1: Left: The launch of Apollo 11 upon the Saturn V rocket. Credit: NASA. Right: Showing the three stages of the Saturn V rocket (each detached once its fuel was expended), plus the Apollo spacecraft on top (containing three astronauts) which was delivered into a translunar orbit. At the base of the rocket is a person to scale, emphasising the enormous size of the rocket. Credit: Encyclopaedia Britannica. | Stage | Initial Mass $(\mathrm{t})$ | Final mass $(\mathrm{t})$ | $I_{\mathrm{sp}}(\mathrm{s})$ | Burn duration $(\mathrm{s})$ | | :---: | :---: | :---: | :---: | :---: | | S-IC | 2283.9 | 135.6 | 263 | 168 | | S-II | 483.7 | 39.9 | 421 | 384 | | S-IV (Burn 1) | 121.0 | - | 421 | 147 | | S-IV (Burn 2) | - | 13.2 | 421 | 347 | | Apollo Spacecraft | 49.7 | - | - | - | Table 1: Data about each stage of the rocket used to launch the Apollo 11 spacecraft into a translunar orbit. Masses are given in tonnes $(1 \mathrm{t}=1000 \mathrm{~kg}$ ) and for convenience include the interstage parts of the rocket too. The specific impulse, $I_{\mathrm{sp}}$, of the stage is given at sea level atmospheric pressure for S-IC and for a vacuum for S-II and S-IVB. The Saturn V rocket consisted of three stages (see Fig 1), since this was the only practical way to get the Apollo spacecraft up to the speed necessary to make the transfer to the Moon. When fully fueled the mass of the total rocket was immense, and lots of that fuel was necessary to simply lift the fuel of the later stages into high altitude - in total about $3000 \mathrm{t}(1$ tonne, $\mathrm{t}=1000 \mathrm{~kg}$ ) of rocket on the launchpad was required to send about $50 \mathrm{t}$ on a mission to the Moon. The first stage (called S-IC) was the heaviest, the second (called S-II) was considerably lighter, and the third stage (called S-IVB) was fired twice - the first to get the spacecraft into a circular 'parking' orbit around the Earth where various safety checks were made, whilst the second burn was to get the spacecraft on its way to the Moon. Once each rocket stage was fully spent it was detached from the rest of the rocket before the next stage ignited. Data about each stage is given in Table 1. The thrust of the rocket is given as $$ F=-I_{\mathrm{sp}} g_{0} \dot{m} $$ where the specific impulse, $I_{\mathrm{sp}}$, of each stage is a constant related to the type of fuel used and the shape of the rocket nozzle, $g_{0}$ is the gravitational field strength of the Earth at sea level (i.e. $g_{0}=9.81 \mathrm{~m} \mathrm{~s}^{-2}$ ) and $\dot{m} \equiv \mathrm{d} m / \mathrm{d} t$ is the rate of change of mass of the rocket with time. The thrust generated by the first two stages (S-IC and S-II) can be taken to be constant. However, the thrust generated by the third stage (S-IVB) varied in order to give a constant acceleration (taken to be the same throughout both burns of the rocket). By the end of the second burn the Apollo spacecraft, apart from a few short burns to give mid-course corrections, coasted all the way to the far side of the Moon where the engines were then fired again to circularise the orbit. All of the early Apollo missions were on a orbit known as a 'free-return trajectory', meaning that if there was a problem then they were already on an orbit that would take them back to Earth after passing around the Moon. The real shape of such a trajectory (in a rotating frame of reference) is like a stretched figure of 8 and is shown in the top panel of Fig 2. To calculate this precisely is non-trivial and required substantial computing power in the 1960s. However, we can have two simplified models that can be used to estimate the duration of the translunar coast, and they are shown in the bottom panel of the Fig 2. The first is a Hohmann transfer orbit (dashed line), which is a single ellipse with the Earth at one focus. In this model the gravitational effect of the Moon is ignored, so the spacecraft travels from A (the perigee) to B (the apogee). The second (solid line) takes advantage of a 'patched conics' approach by having two ellipses whose apoapsides coincide at point $\mathrm{C}$ where the gravitational force on the spacecraft is equal from both the Earth and the Moon. The first ellipse has a periapsis at A and ignores the gravitational effect of the Moon, whilst the second ellipse has a periapsis at B and ignores the gravitational effect of the Earth. If the spacecraft trajectory and lunar orbit are coplanar and the Moon is in a circular orbit around the Earth then the time to travel from $\mathrm{A}$ to $\mathrm{B}$ via $\mathrm{C}$ is double the value attained if taking into account the gravitational forces of the Earth and Moon together throughout the journey, which is a much better estimate of the time of a real translunar coast. [figure2] Figure 2: Top: The real shape of a translunar free-return trajectory, with the Earth on the left and the Moon on the right (orbiting around the Earth in an anti-clockwise direction). This diagram (and the one below) is shown in a co-ordinate system co-rotating with the Earth and is not to scale. Credit: NASA. Bottom: Two simplified ways of modelling the translunar trajectory. The simplest is a Hohmann transfer orbit (dashed line, outer ellipse), which is an ellipse that has the Earth at one focus and ignores the gravitational effect of the Moon. A better model (solid line, inner ellipses) of the Apollo trajectory is the use of two ellipses that meet at point $\mathrm{C}$ where the gravitational forces of the Earth and Moon on the spacecraft are equal. For the Apollo 11 journey, the end of the second burn of the S-IVB rocket (point A) was $334 \mathrm{~km}$ above the surface of the Earth, and the end of the translunar coast (point B) was $161 \mathrm{~km}$ above the surface of the Moon. The distance between the centres of mass of the Earth and the Moon at the end of the translunar coast was $3.94 \times 10^{8} \mathrm{~m}$. Take the radius of the Earth to be $6370 \mathrm{~km}$, the radius of the Moon to be $1740 \mathrm{~km}$, and the mass of the Moon to be $7.35 \times 10^{22} \mathrm{~kg}$. problem: d. For the patched conics approach (solid lines): ii. Calculate the speed of the spacecraft at point A and point B. Give your answer in $\mathrm{km} \mathrm{s}^{-1}$ and as a percentage of the escape speed of the spacecraft at that distance from the relevant closest gravitational body. Comment on what this implies for the eccentricity of the orbits. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of \% v_{\mathrm{esc}, B}, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

multi-modal

Astronomy_454

地球刚诞生时自转周期约是 8 小时, 因为受到月球潮汐的影响, 自转在持续减速,现在地球自转周期是 24 小时。与此同时, 在数年、数十年的时间内, 由于地球板块的运动、地壳的收缩、海洋、大气等一些复杂因素以及人类活动的影响, 地球的自转周期会发生毫秒级别的微小波动。科学研究指出, 若不考虑月球的影响, 在地球的总质量不变的情况下, 地球上的所有物质满足 $m_{1} \omega r_{1}{ }^{2}+m_{2} \omega r_{2}{ }^{2}+\ldots+m_{i} \omega r_{i}{ }^{2}=$ 常量, 其中 $m_{1}$ 、 $m_{2} 、 \ldots \ldots m_{i}$ 表示地球各部分的质量, $r_{1} 、 r_{2} 、 \ldots \ldots r_{i}$ 为地球各部分到地轴的距离, $\omega$ 为地球自转的角速度, 如图所示。根据以上信息, 结合所学, 判断下列说法正确的是 ( ) [图1] A: 月球潮汐的影响使地球自转的角速度变大 B: 若地球自转变慢, 地球赤道处的重力加速度会变小 C: 若仅考虑 $A$ 处的冰川融化, 质心下降, 会使地球自转周期变小 D: 若仅考虑 $B$ 处板块向赤道漂移，会使地球自转周期变小

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 地球刚诞生时自转周期约是 8 小时, 因为受到月球潮汐的影响, 自转在持续减速,现在地球自转周期是 24 小时。与此同时, 在数年、数十年的时间内, 由于地球板块的运动、地壳的收缩、海洋、大气等一些复杂因素以及人类活动的影响, 地球的自转周期会发生毫秒级别的微小波动。科学研究指出, 若不考虑月球的影响, 在地球的总质量不变的情况下, 地球上的所有物质满足 $m_{1} \omega r_{1}{ }^{2}+m_{2} \omega r_{2}{ }^{2}+\ldots+m_{i} \omega r_{i}{ }^{2}=$ 常量, 其中 $m_{1}$ 、 $m_{2} 、 \ldots \ldots m_{i}$ 表示地球各部分的质量, $r_{1} 、 r_{2} 、 \ldots \ldots r_{i}$ 为地球各部分到地轴的距离, $\omega$ 为地球自转的角速度, 如图所示。根据以上信息, 结合所学, 判断下列说法正确的是 ( ) [图1] A: 月球潮汐的影响使地球自转的角速度变大 B: 若地球自转变慢, 地球赤道处的重力加速度会变小 C: 若仅考虑 $A$ 处的冰川融化, 质心下降, 会使地球自转周期变小 D: 若仅考虑 $B$ 处板块向赤道漂移，会使地球自转周期变小 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

multi-modal

Astronomy_564

2020 年 7 月 23 日, 我国“天问一号”火星探测器成功发射, 2021 年 2 月 10 日, 顺利进入环火星大椭圆轨道, 并变轨到近火星圆轨道运动, 将于 2021 年 5 月至 6 月择机实施火星着陆, 最终实现“绕、着、巡”三大目标。已知火星质量约为地球的 $\frac{1}{10}$, 半径约为地球的 $\frac{1}{2}$, 地球表面的重力加速度为 $g$, 火星和地球均绕太阳做逆时针方向的匀速圆周运动, 火星的公转周期是地球公转周期的两倍。质量为 $m$ 的着陆器在着陆火星前,会在火星表面附近经历一个时长为 $t$ ，速度由 $v$ 减速到零的过程. 若该减速过程可以视为一个坚直向下的匀减速直线运动, 忽略火星大气阻力, 求: 着陆过程中，着陆器受到的制动力大小;

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 2020 年 7 月 23 日, 我国“天问一号”火星探测器成功发射, 2021 年 2 月 10 日, 顺利进入环火星大椭圆轨道, 并变轨到近火星圆轨道运动, 将于 2021 年 5 月至 6 月择机实施火星着陆, 最终实现“绕、着、巡”三大目标。已知火星质量约为地球的 $\frac{1}{10}$, 半径约为地球的 $\frac{1}{2}$, 地球表面的重力加速度为 $g$, 火星和地球均绕太阳做逆时针方向的匀速圆周运动, 火星的公转周期是地球公转周期的两倍。质量为 $m$ 的着陆器在着陆火星前,会在火星表面附近经历一个时长为 $t$ ，速度由 $v$ 减速到零的过程. 若该减速过程可以视为一个坚直向下的匀减速直线运动, 忽略火星大气阻力, 求: 着陆过程中，着陆器受到的制动力大小; 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

EX

Astronomy

ZH

text-only

Astronomy_335

“神九”载人飞船与“天宫一号”成功对接及“蛟龙”号下潜突破 7000 米入选 2012 年中国十大科技进展新闻。若地球半径为 $R$, 把地球看作质量分布均匀的球体（质量分布均匀的球壳对球内任一质点的万有引力为零)。“蛟龙”号下潜深度为 $d$, “天宫一号”轨道距离地面高度为 $h$, “天宫一号”所在处与“蛟龙”号所在处的重力加速度之比为 ( ) A: $\frac{R-d}{R+h}$ B: $\frac{R^{3}}{(R-d)(R+h)^{2}}$ C: $\frac{(R-d)(R+h)^{2}}{R^{3}}$ D: $\frac{(R-d)(R+h)}{R^{2}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: “神九”载人飞船与“天宫一号”成功对接及“蛟龙”号下潜突破 7000 米入选 2012 年中国十大科技进展新闻。若地球半径为 $R$, 把地球看作质量分布均匀的球体（质量分布均匀的球壳对球内任一质点的万有引力为零)。“蛟龙”号下潜深度为 $d$, “天宫一号”轨道距离地面高度为 $h$, “天宫一号”所在处与“蛟龙”号所在处的重力加速度之比为 ( ) A: $\frac{R-d}{R+h}$ B: $\frac{R^{3}}{(R-d)(R+h)^{2}}$ C: $\frac{(R-d)(R+h)^{2}}{R^{3}}$ D: $\frac{(R-d)(R+h)}{R^{2}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

text-only

Astronomy_17

已知地球半径为 $R$, 地球表面重力加速度为 $g$, 卫星绕地球运动轨道为圆轨道, 在某时刻地球的两颗卫星 $\mathrm{A}$ 和 $\mathrm{B}$ 处于如图所示的位置, 设卫星 $\mathrm{A}$ 的轨道半径为 $r_{A}=2 R$,卫星 $\mathrm{B}$ 的轨道半径为 $r_{B}=3 R$, [已知 $\frac{12}{19}(3 \sqrt{3}+2 \sqrt{2}) \approx 5$ ], 则下列说法正确的是 ( ) [图1] A: 两卫星的环绕速度之比 $\frac{v_{A}}{v_{B}}=\frac{\sqrt{3}}{\sqrt{2}}$ B: 开始时, 卫星 A 将在 B 的前面, 只要 B 加速即可追上 A C: 从此时刻开始再经大约 $5 \pi \sqrt{\frac{6 R}{g}}$ 时间再次相距最近 D: 从此时刻开始再经大约 $\frac{3}{4} \pi \sqrt{\frac{2 R}{g}}$ 时间第一次相距最远

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 已知地球半径为 $R$, 地球表面重力加速度为 $g$, 卫星绕地球运动轨道为圆轨道, 在某时刻地球的两颗卫星 $\mathrm{A}$ 和 $\mathrm{B}$ 处于如图所示的位置, 设卫星 $\mathrm{A}$ 的轨道半径为 $r_{A}=2 R$,卫星 $\mathrm{B}$ 的轨道半径为 $r_{B}=3 R$, [已知 $\frac{12}{19}(3 \sqrt{3}+2 \sqrt{2}) \approx 5$ ], 则下列说法正确的是 ( ) [图1] A: 两卫星的环绕速度之比 $\frac{v_{A}}{v_{B}}=\frac{\sqrt{3}}{\sqrt{2}}$ B: 开始时, 卫星 A 将在 B 的前面, 只要 B 加速即可追上 A C: 从此时刻开始再经大约 $5 \pi \sqrt{\frac{6 R}{g}}$ 时间再次相距最近 D: 从此时刻开始再经大约 $\frac{3}{4} \pi \sqrt{\frac{2 R}{g}}$ 时间第一次相距最远 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_448

某兴趣小组想在地球建造从地表直达地心的隧道。若将地球视为质量分布均匀的标准球体，质量为 $M$, 半径为 $R$ 。已知质量分布均匀的球壳对内部引力处处为零，万有引力常量为 $G$, 忽略地球自转。则沿该隧道从地表静止释放的物体, 到达地心处的速度为 A: $\sqrt{\frac{G M}{R}}$ B: $\frac{1}{2} \sqrt{\frac{G M}{R}}$ C: $\sqrt{\frac{G M}{2 R}}$ D: $\sqrt{\frac{2 G M}{R}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 某兴趣小组想在地球建造从地表直达地心的隧道。若将地球视为质量分布均匀的标准球体，质量为 $M$, 半径为 $R$ 。已知质量分布均匀的球壳对内部引力处处为零，万有引力常量为 $G$, 忽略地球自转。则沿该隧道从地表静止释放的物体, 到达地心处的速度为 A: $\sqrt{\frac{G M}{R}}$ B: $\frac{1}{2} \sqrt{\frac{G M}{R}}$ C: $\sqrt{\frac{G M}{2 R}}$ D: $\sqrt{\frac{2 G M}{R}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

text-only

Astronomy_605

宇宙空间有两颗相距较远、中心距离为 $d$ 的星球 $\mathrm{A}$ 和星球 $\mathrm{B}$ 。在星球 $\mathrm{A}$ 上将一轻弹簧坚直固定在水平桌面上, 把物体 P 轻放在弹簧上端, 如图 (a) 所示, P 由静止向下运动, 其加速度 $a$ 与弹簧的压缩量 $x$ 间的关系如图 (b) 中实线所示。在星球 $\mathrm{B}$ 上用完全相同的弹簧和物体 $\mathrm{P}$ 完成同样的过程，其 $a-x$ 关系如图（b）中虚线所示。已知两星 球密度相等。星球 $\mathrm{A}$ 的质量为 $m_{0}$, 引力常量为 $G$ 。假设两星球均为质量均匀分布的球体。则下列判断正确的是 ( ) [图1] 图(a) [图2] 图(b) A: 星球 B 的质量为 $8 m_{0}$ B: 星球 $\mathrm{A}$ 和星球 B 的表面重力加速度的比值为 2 C: 若星球 $\mathrm{A}$ 绕星球 $\mathrm{B}$ 做匀速圆周运动, 则星球 $\mathrm{A}$ 的运行周期 $T_{1}=\pi d \cdot \sqrt{\frac{d}{2 G m_{0}}}$ D: 若将星球 $\mathrm{A} 、 \mathrm{~B}$ 看成是远离其他星球的双星模型, 则它们的周期 $T_{2}=\pi d \cdot \sqrt{\frac{d}{G m_{0}}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 宇宙空间有两颗相距较远、中心距离为 $d$ 的星球 $\mathrm{A}$ 和星球 $\mathrm{B}$ 。在星球 $\mathrm{A}$ 上将一轻弹簧坚直固定在水平桌面上, 把物体 P 轻放在弹簧上端, 如图 (a) 所示, P 由静止向下运动, 其加速度 $a$ 与弹簧的压缩量 $x$ 间的关系如图 (b) 中实线所示。在星球 $\mathrm{B}$ 上用完全相同的弹簧和物体 $\mathrm{P}$ 完成同样的过程，其 $a-x$ 关系如图（b）中虚线所示。已知两星 球密度相等。星球 $\mathrm{A}$ 的质量为 $m_{0}$, 引力常量为 $G$ 。假设两星球均为质量均匀分布的球体。则下列判断正确的是 ( ) [图1] 图(a) [图2] 图(b) A: 星球 B 的质量为 $8 m_{0}$ B: 星球 $\mathrm{A}$ 和星球 B 的表面重力加速度的比值为 2 C: 若星球 $\mathrm{A}$ 绕星球 $\mathrm{B}$ 做匀速圆周运动, 则星球 $\mathrm{A}$ 的运行周期 $T_{1}=\pi d \cdot \sqrt{\frac{d}{2 G m_{0}}}$ D: 若将星球 $\mathrm{A} 、 \mathrm{~B}$ 看成是远离其他星球的双星模型, 则它们的周期 $T_{2}=\pi d \cdot \sqrt{\frac{d}{G m_{0}}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_44

所谓“双星系统”, 是指在相互间引力的作用下, 绕连线上某点 $O$ 做匀速圆周运动的两个星球 $\mathrm{A}$ 和 $\mathrm{B}$, 如图所示, 星球 $\mathrm{A} 、 \mathrm{~B}$ 绕 $O$ 点做匀速圆周运动的周期为 $T$, 其中 $\mathrm{A}$星球表面重力加速度为 $g$, 半径为 $R$, $\mathrm{A}$ 星球的自转忽略不计, $\mathrm{B}$ 星球的质量为 $m$, 引力常量为 $G$, 则 $\mathrm{A} 、 \mathrm{~B}$ 两星球间的距离 $L$ 可表示为 ( ) [图1] A: $\sqrt[3]{\frac{\left(g R^{2}+G m\right) T^{2}}{4 \pi^{2}}}$ B: $\sqrt{\frac{\left(g R^{2}+G m\right) T^{2}}{4 \pi^{2}}}$ C: $\sqrt[3]{\frac{g R^{2} T^{2}}{4 \pi^{2}}}$ D: $\sqrt{\frac{m g R^{2} T^{2}}{4 G \pi^{2}}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 所谓“双星系统”, 是指在相互间引力的作用下, 绕连线上某点 $O$ 做匀速圆周运动的两个星球 $\mathrm{A}$ 和 $\mathrm{B}$, 如图所示, 星球 $\mathrm{A} 、 \mathrm{~B}$ 绕 $O$ 点做匀速圆周运动的周期为 $T$, 其中 $\mathrm{A}$星球表面重力加速度为 $g$, 半径为 $R$, $\mathrm{A}$ 星球的自转忽略不计, $\mathrm{B}$ 星球的质量为 $m$, 引力常量为 $G$, 则 $\mathrm{A} 、 \mathrm{~B}$ 两星球间的距离 $L$ 可表示为 ( ) [图1] A: $\sqrt[3]{\frac{\left(g R^{2}+G m\right) T^{2}}{4 \pi^{2}}}$ B: $\sqrt{\frac{\left(g R^{2}+G m\right) T^{2}}{4 \pi^{2}}}$ C: $\sqrt[3]{\frac{g R^{2} T^{2}}{4 \pi^{2}}}$ D: $\sqrt{\frac{m g R^{2} T^{2}}{4 G \pi^{2}}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

multi-modal

Astronomy_302

2023 年 11 月 18 日出现了“火星合日”现象, 即当火星和地球分别位于太阳两侧与太阳共线干扰无线电时, 影响通信的天文现象, 因此中国首辆火星车“祝融号”（在火星赤道表面附近做匀速圆周运动）发生短暂“失联”。如图所示, 已知地球与火星绕太阳做匀速圆周运动, 且运动方向相同, 火星、地球公转轨道半径之比为 $3: 2$, 忽略行星间的作用力。下列说法正确的是 ( ) [图1] A: 火星、地球绕太阳公转的线速度之比为 $\sqrt{2}: \sqrt{3}$ B: 火星、地球绕太阳公转的周期之比为 $\sqrt[3]{9}: \sqrt[3]{4}$ C: 相同时间内, 火星与太阳连线、地球与太阳连线扫过的面积之比为 $\sqrt{3}: \sqrt{2}$ D: 下一次“火星合日”将出现在 2025 年 11 月 18 日之后

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 2023 年 11 月 18 日出现了“火星合日”现象, 即当火星和地球分别位于太阳两侧与太阳共线干扰无线电时, 影响通信的天文现象, 因此中国首辆火星车“祝融号”（在火星赤道表面附近做匀速圆周运动）发生短暂“失联”。如图所示, 已知地球与火星绕太阳做匀速圆周运动, 且运动方向相同, 火星、地球公转轨道半径之比为 $3: 2$, 忽略行星间的作用力。下列说法正确的是 ( ) [图1] A: 火星、地球绕太阳公转的线速度之比为 $\sqrt{2}: \sqrt{3}$ B: 火星、地球绕太阳公转的周期之比为 $\sqrt[3]{9}: \sqrt[3]{4}$ C: 相同时间内, 火星与太阳连线、地球与太阳连线扫过的面积之比为 $\sqrt{3}: \sqrt{2}$ D: 下一次“火星合日”将出现在 2025 年 11 月 18 日之后 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

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Astronomy

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multi-modal

Astronomy_428

2020 年 5 月 24 日, 中国航天科技集团发文表示, 我国正按计划推进火星探测工程, 瞄准今年 7 月将火星探测器发射升空。假设探测器贴近火星地面做匀速圆周运动时, 绕行周期为 $T$, 已知火星半径为 $R$, 万有引力常量为 $G$, 由此可以估算 ( ) A: 火星质量 B: 探测器质量 C: 火星第一宇宙速度 D: 火星平均密度

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 2020 年 5 月 24 日, 中国航天科技集团发文表示, 我国正按计划推进火星探测工程, 瞄准今年 7 月将火星探测器发射升空。假设探测器贴近火星地面做匀速圆周运动时, 绕行周期为 $T$, 已知火星半径为 $R$, 万有引力常量为 $G$, 由此可以估算 ( ) A: 火星质量 B: 探测器质量 C: 火星第一宇宙速度 D: 火星平均密度 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

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Astronomy

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text-only

Astronomy_506

牛顿在发现万有引力定律后曾思考过这样一个问题: 假设地球是一个质量为 $M$,半径为 $R$, 质量均匀分布的球体, 已知质量分布均匀的球壳对球壳内物体的引力为零。沿地球的南北极打一个内壁光滑的洞，在洞口无初速度释放一个质量为 $m$ 的小球 (小球的直径略小于洞的直径), 在小球向下端点运动的过程中, 已知万有引力常量为 $G$ 。下列说法中正确的是 ( ) A: 小球运动到洞的下端点所用的时间为 $\frac{\pi}{2} \sqrt{\frac{R^{3}}{G M}}$ B: 小球在距地心 $x$ 处所受到的万有引力的合力大小为 $F=\frac{G M m x}{R^{3}}$ C: 小球在洞内做往复振动, 小球受力与到地心距离的关系决定了此振动为非简谐运动 D: 若小球释放的位置再向下移动一点, 则小球振动周期变小

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 牛顿在发现万有引力定律后曾思考过这样一个问题: 假设地球是一个质量为 $M$,半径为 $R$, 质量均匀分布的球体, 已知质量分布均匀的球壳对球壳内物体的引力为零。沿地球的南北极打一个内壁光滑的洞，在洞口无初速度释放一个质量为 $m$ 的小球 (小球的直径略小于洞的直径), 在小球向下端点运动的过程中, 已知万有引力常量为 $G$ 。下列说法中正确的是 ( ) A: 小球运动到洞的下端点所用的时间为 $\frac{\pi}{2} \sqrt{\frac{R^{3}}{G M}}$ B: 小球在距地心 $x$ 处所受到的万有引力的合力大小为 $F=\frac{G M m x}{R^{3}}$ C: 小球在洞内做往复振动, 小球受力与到地心距离的关系决定了此振动为非简谐运动 D: 若小球释放的位置再向下移动一点, 则小球振动周期变小 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

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Astronomy_708

中国计划已经实现返回式月球软着陆器对月球进行科学探测, 如图所示, 发射一颗运动半径为 $r$ 的绕月卫星, 登月着陆器从绕月卫星出发 (不影响绕月卫星运动), 沿粗圆轨道降落到月球的表面上, 与月球表面经多次碰撞和弹跳才停下来。假设着陆器第一次弹起的最大高度为 $h$, 水平速度为 $v_{1}$, 第二次着陆时速度为 $v_{2}$, 已知月球半径为 $R$,着陆器质量为 $m$, 不计一切阻力和月球的自转。求: 月球表面的重力加速度 $g_{\text {月。 }}$ [图1]

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个判断题（对或错）。 问题: 中国计划已经实现返回式月球软着陆器对月球进行科学探测, 如图所示, 发射一颗运动半径为 $r$ 的绕月卫星, 登月着陆器从绕月卫星出发 (不影响绕月卫星运动), 沿粗圆轨道降落到月球的表面上, 与月球表面经多次碰撞和弹跳才停下来。假设着陆器第一次弹起的最大高度为 $h$, 水平速度为 $v_{1}$, 第二次着陆时速度为 $v_{2}$, 已知月球半径为 $R$,着陆器质量为 $m$, 不计一切阻力和月球的自转。求: 月球表面的重力加速度 $g_{\text {月。 }}$ [图1] 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为“True”或“False”。

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Astronomy

ZH

multi-modal

Astronomy_611

$P_{1} 、 P_{2}$ 为相距遥远的两颗行星, 距各自表面相同高度处各有一颗卫星 $s_{1} 、 s_{2}$ 做匀速圆周运动, 图中纵坐标表示行星对周围空间各处物体的引力产生的加速度 $a$, 横坐标表示物体到行星中心的距离 $r$ 的平方, 两条曲线分别表示 $P_{1} 、 P_{2}$ 周围的 $a$ 与 $r^{2}$ 的反比关系,它们左端点横坐标相同，则( ) [图1] A: $P_{1} 、 P_{2}$ 的平均密度相等 B: $P_{1}$ 的第一宇宙速度比 $P_{2}$ 的小 C: $s_{1}$ 的公转周期比 $s_{2}$ 的大 D: $s_{1}$ 的向心加速度比 $s_{2}$ 的大

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: $P_{1} 、 P_{2}$ 为相距遥远的两颗行星, 距各自表面相同高度处各有一颗卫星 $s_{1} 、 s_{2}$ 做匀速圆周运动, 图中纵坐标表示行星对周围空间各处物体的引力产生的加速度 $a$, 横坐标表示物体到行星中心的距离 $r$ 的平方, 两条曲线分别表示 $P_{1} 、 P_{2}$ 周围的 $a$ 与 $r^{2}$ 的反比关系,它们左端点横坐标相同，则( ) [图1] A: $P_{1} 、 P_{2}$ 的平均密度相等 B: $P_{1}$ 的第一宇宙速度比 $P_{2}$ 的小 C: $s_{1}$ 的公转周期比 $s_{2}$ 的大 D: $s_{1}$ 的向心加速度比 $s_{2}$ 的大 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

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multi-modal

Astronomy_163

两颗互不影响的行星 $P_{1} 、 P_{2}$, 各有一颗在表面附近的卫星 $S_{1} 、 S_{2}$ 绕其做匀速圆周运动。两颗行星周围卫星的线速度的二次方 $\left(v^{2}\right)$ 与轨道半径 $r$ 的倒数 $\left(\frac{1}{r}\right)$ 的关系如图所示, 已知 $S_{1} 、 S_{2}$ 的线速度大小均为 $v_{0}$, 则 ( ) [图1] A: $S_{1}$ 的质量比 $S_{2}$ 的小 B: $P_{1}$ 的质量比 $P_{2}$ 的小 C: $P_{1}$ 的平均密度比 $P_{2}$ 的小 D: $P_{1}$ 表面的重力加速度比 $P_{2}$ 的小

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 两颗互不影响的行星 $P_{1} 、 P_{2}$, 各有一颗在表面附近的卫星 $S_{1} 、 S_{2}$ 绕其做匀速圆周运动。两颗行星周围卫星的线速度的二次方 $\left(v^{2}\right)$ 与轨道半径 $r$ 的倒数 $\left(\frac{1}{r}\right)$ 的关系如图所示, 已知 $S_{1} 、 S_{2}$ 的线速度大小均为 $v_{0}$, 则 ( ) [图1] A: $S_{1}$ 的质量比 $S_{2}$ 的小 B: $P_{1}$ 的质量比 $P_{2}$ 的小 C: $P_{1}$ 的平均密度比 $P_{2}$ 的小 D: $P_{1}$ 表面的重力加速度比 $P_{2}$ 的小 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

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Astronomy

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multi-modal

Astronomy_80

有 $a 、 b 、 c 、 d$ 四颗地球卫星： $a$ 还未发射, 在地球赤道上随地球表面一起转动; $b$ 处于离地很近的近地圆轨道上正常运动; $c$ 是地球同步卫星; $d$是高空探测卫星. 各卫星排列位置如图, 则下列说法正确的是( ) [图1] A: $a$ 的向心加速度等于重力加速度 $g$ B: 把 $a$ 直接发射到 $b$ 运行的轨道上, 其发射速度大于第一宇宙速度 C: $b$ 在相同时间内转过的弧长最长 D: $d$ 的运动周期有可能是 $20 \mathrm{~h}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 有 $a 、 b 、 c 、 d$ 四颗地球卫星： $a$ 还未发射, 在地球赤道上随地球表面一起转动; $b$ 处于离地很近的近地圆轨道上正常运动; $c$ 是地球同步卫星; $d$是高空探测卫星. 各卫星排列位置如图, 则下列说法正确的是( ) [图1] A: $a$ 的向心加速度等于重力加速度 $g$ B: 把 $a$ 直接发射到 $b$ 运行的轨道上, 其发射速度大于第一宇宙速度 C: $b$ 在相同时间内转过的弧长最长 D: $d$ 的运动周期有可能是 $20 \mathrm{~h}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

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multi-modal

Astronomy_499

$P_{1}, P_{2}$ 为相距遥远的两颗行星, 距各自表面相同高度处各有一颗卫星 $s_{1}, s_{2}$ 做匀速圆周运动. 如图所示, 纵坐标表示行星对其周围空间各处物体的引力产生的加速度 $a$,横坐标表示物体到行星中心的距离 $r$ 的平方, 两条曲线分别表示 $P_{1}, P_{2}$ 周围的 $a$ 的大小与 $r^{2}$ 的反比关系, 它们的左端点横坐标相同. 则( ) [图1] A: $P_{1}$ 的“第一宇宙速度”比 $P_{2}$ 的小 B: $P_{1}$ 的平均密度比 $P_{2}$ 的大 C: $s_{1}$ 的向心加速度比 $s_{2}$ 的大 D: $s_{1}$ 的公转周期比 $s_{2}$ 的大

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: $P_{1}, P_{2}$ 为相距遥远的两颗行星, 距各自表面相同高度处各有一颗卫星 $s_{1}, s_{2}$ 做匀速圆周运动. 如图所示, 纵坐标表示行星对其周围空间各处物体的引力产生的加速度 $a$,横坐标表示物体到行星中心的距离 $r$ 的平方, 两条曲线分别表示 $P_{1}, P_{2}$ 周围的 $a$ 的大小与 $r^{2}$ 的反比关系, 它们的左端点横坐标相同. 则( ) [图1] A: $P_{1}$ 的“第一宇宙速度”比 $P_{2}$ 的小 B: $P_{1}$ 的平均密度比 $P_{2}$ 的大 C: $s_{1}$ 的向心加速度比 $s_{2}$ 的大 D: $s_{1}$ 的公转周期比 $s_{2}$ 的大 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

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Astronomy

ZH

multi-modal

Astronomy_238

嫦娥一号月球探测器发射时在绕地球运行中进行了四次变轨, 其中有一次变轨是提高近地点的高度, 使之从距地 $200 \mathrm{~km}$, 上升到距地 $600 \mathrm{~km}$, 这样既提高了飞船飞行高度,又减缓飞船经过近地点的速度, 增长测控时间, 关于这次变轨说法正确的是 ( ) A: 变轨后探测器在远地点的加速度变大 B: 应在近地点向运动后方喷气 C: 应在远地点向运动后方喷气 D: 变轨后探测器的周期将变小

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 嫦娥一号月球探测器发射时在绕地球运行中进行了四次变轨, 其中有一次变轨是提高近地点的高度, 使之从距地 $200 \mathrm{~km}$, 上升到距地 $600 \mathrm{~km}$, 这样既提高了飞船飞行高度,又减缓飞船经过近地点的速度, 增长测控时间, 关于这次变轨说法正确的是 ( ) A: 变轨后探测器在远地点的加速度变大 B: 应在近地点向运动后方喷气 C: 应在远地点向运动后方喷气 D: 变轨后探测器的周期将变小 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy_659

“天问一号”已于 2020 年 7 月 23 日在中国文昌航天发射场由长征五号遥四运载火箭发射升空，成功进入预定轨道。“天问一号”将完成“环绕”“着陆”“巡视”火星这三大任务。 已知日地间距约为 1.5 亿公里, 火星直径约为地球的一半, 质量约为地球的 $11 \%$, 将火星和地球绕太阳的运动均视为圆周运动, 两者每隔约 2.2 年相遇（相距最近）一次, 不考虑火星和地球间的万有引力，地球公转周期视为 1 年。下列说法正确的是 A: “天问一号”的发射速度必须大于地球第一宇宙速度且小于第二宇宙速度 B: 火星表面的“重力加速度”大于地球表面的重力加速度 C: 由上述材料和天文学常识可以估算出火星公转的周期 D: 由上述材料和天文学常识可以估算出火星的密度

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: “天问一号”已于 2020 年 7 月 23 日在中国文昌航天发射场由长征五号遥四运载火箭发射升空，成功进入预定轨道。“天问一号”将完成“环绕”“着陆”“巡视”火星这三大任务。 已知日地间距约为 1.5 亿公里, 火星直径约为地球的一半, 质量约为地球的 $11 \%$, 将火星和地球绕太阳的运动均视为圆周运动, 两者每隔约 2.2 年相遇（相距最近）一次, 不考虑火星和地球间的万有引力，地球公转周期视为 1 年。下列说法正确的是 A: “天问一号”的发射速度必须大于地球第一宇宙速度且小于第二宇宙速度 B: 火星表面的“重力加速度”大于地球表面的重力加速度 C: 由上述材料和天文学常识可以估算出火星公转的周期 D: 由上述材料和天文学常识可以估算出火星的密度 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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text-only

Astronomy_602

一宇航员到达半径为 $R$, 密度均匀的某星球表面, 做如下实验, 用不可伸长的轻绳拴一质量为 $m$ 的小球, 上端固定为 $O$ 点, 如图所示, 在最低点给小球某一初速度, 使其绕 $O$ 点在坚直面内做圆周运动, 测得绳的拉力大小 $F$ 随时间 $t$ 的变化规律。如图乙所示, 若 $F_{1}$ 的大小等于 $7 F_{2}$, 且设 $R 、 m$ 、引力常量为 $G 、 F_{1}$ 为已知量, 忽略各种阻力,则以下说法正确的是（ ） [图1] 甲 [图2] 乙 A: 该星球表面的重力加速度为 $\frac{F_{1}}{7 m}$ B: 卫星绕该星球的第一宇宙速度为 $\sqrt{\frac{G \pi}{R}}$ C: 星球的质量为 $\frac{F_{1} R^{2}}{7 G m}$ D: 小球通过最高点的最小速度为 $\sqrt{g R}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 一宇航员到达半径为 $R$, 密度均匀的某星球表面, 做如下实验, 用不可伸长的轻绳拴一质量为 $m$ 的小球, 上端固定为 $O$ 点, 如图所示, 在最低点给小球某一初速度, 使其绕 $O$ 点在坚直面内做圆周运动, 测得绳的拉力大小 $F$ 随时间 $t$ 的变化规律。如图乙所示, 若 $F_{1}$ 的大小等于 $7 F_{2}$, 且设 $R 、 m$ 、引力常量为 $G 、 F_{1}$ 为已知量, 忽略各种阻力,则以下说法正确的是（ ） [图1] 甲 [图2] 乙 A: 该星球表面的重力加速度为 $\frac{F_{1}}{7 m}$ B: 卫星绕该星球的第一宇宙速度为 $\sqrt{\frac{G \pi}{R}}$ C: 星球的质量为 $\frac{F_{1} R^{2}}{7 G m}$ D: 小球通过最高点的最小速度为 $\sqrt{g R}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_693

太阳系中, 行星周围存在着“作用球”空间: 在该空间内, 探测器的运动特征主要决定于行星的引力。 2020 年中国将首次发射火星探测器, 并一次实现“环绕、着陆、巡视”三个目标。如图所示, 若将火星探测器的发射过程简化为以下三个阶段: 在地心轨道沿地球作用球边界飞行, 进入日心转移轨道环绕太阳飞行, 在俘获轨道沿火星作用球边界飞行。且 $A$ 点为地心轨道与日心转移轨道切点, $B$ 点为日心转移轨道与俘获轨道切点,则下列关于火星探测器说法正确的是（） [图1] A: 在地心轨道上经过 $A$ 点的速度小于在日心转移轨道上经过 $A$ 点的速度 B: 在 $B$ 点受到火星对它的引力大于太阳对它的引力 C: 在 $C$ 点的运行速率大于地球的公转速率 D: 若已知其在俘获轨道运行周期, 可估算火星密度

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 太阳系中, 行星周围存在着“作用球”空间: 在该空间内, 探测器的运动特征主要决定于行星的引力。 2020 年中国将首次发射火星探测器, 并一次实现“环绕、着陆、巡视”三个目标。如图所示, 若将火星探测器的发射过程简化为以下三个阶段: 在地心轨道沿地球作用球边界飞行, 进入日心转移轨道环绕太阳飞行, 在俘获轨道沿火星作用球边界飞行。且 $A$ 点为地心轨道与日心转移轨道切点, $B$ 点为日心转移轨道与俘获轨道切点,则下列关于火星探测器说法正确的是（） [图1] A: 在地心轨道上经过 $A$ 点的速度小于在日心转移轨道上经过 $A$ 点的速度 B: 在 $B$ 点受到火星对它的引力大于太阳对它的引力 C: 在 $C$ 点的运行速率大于地球的公转速率 D: 若已知其在俘获轨道运行周期, 可估算火星密度 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_549

图中圆周是地球绕太阳运动的轨道, 地球处在 $b$ 位置, 地球和太阳连线上的 $a$ 与 $e$位置、 $c$ 与 $d$ 位置均关于太阳对称, 当一无动力的探测器处在 $a$ 或 $c$ 位置时, 它仅在太阳和地球引力的共同作用下，与地球一起以相同的周期绕太阳做圆周运动，下列分析正确的是 ( ) [图1] A: 该探测器在 $a$ 位置的线速度大于 $c$ 位置的线速度 B: 该探测器在 $a$ 位置受到的引力大于 $c$ 位置受到的引力 C: 若地球和该探测器分别在 $b 、 d$ 位置，它们也能以相同的周期运动 D: 若地球和该探测器分别在 $b 、 e$ 位置，它们也能以相同的周期运动

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 图中圆周是地球绕太阳运动的轨道, 地球处在 $b$ 位置, 地球和太阳连线上的 $a$ 与 $e$位置、 $c$ 与 $d$ 位置均关于太阳对称, 当一无动力的探测器处在 $a$ 或 $c$ 位置时, 它仅在太阳和地球引力的共同作用下，与地球一起以相同的周期绕太阳做圆周运动，下列分析正确的是 ( ) [图1] A: 该探测器在 $a$ 位置的线速度大于 $c$ 位置的线速度 B: 该探测器在 $a$ 位置受到的引力大于 $c$ 位置受到的引力 C: 若地球和该探测器分别在 $b 、 d$ 位置，它们也能以相同的周期运动 D: 若地球和该探测器分别在 $b 、 e$ 位置，它们也能以相同的周期运动 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_1106

Young, Earth-like planets interact with the protoplanetary discs in which they form, and as a result migrate to different orbital radii. The aim of this question is to quantify this migration in a simple model. We shall think of protoplanetary discs as consisting of nested circular orbits of gas and dust around a central star. For a thin disc (with small vertical extent), we assign to the disc a surface density (or mass per unit area) $\Sigma$, and semi-thickness $H$, which in general vary over the disc's extent. The disc's 'aspect ratio' at radius $r$ from the central star is denoted $h=H / r$. This question is concerned with the migration of 'small' planets, such that $M_{p} / M_{\star}=q \ll h^{3}$. [figure1] Figure 6: Left: ALMA image of the young star HL Tau and its protoplanetary disk. This image of planet formation reveals multiple rings and gaps that herald the presence of emerging planets as they sweep their orbits clear of dust and gas. Credit: ALMA (NRAO/ESO/NAOJ) / C. Brogan, B. Saxton (NRAO/AUI/NSF). Right: A small planet orbits whilst embedded in a protoplanetary disc, exciting a 1-armed spiral density wave. Credit: Frdric Masset. Since the planet is assumed small $\left(q \ll h^{3}\right)$, its interaction with the gas in the disc constitutes the excitation of a spiral density wave, and redistribution of matter in the co-orbital region (that is, matter orbiting at radii $r \approx r_{p}$ ), as shown in Figure 6. The resulting non-uniform density distribution induced in the disc exerts a net gravitational force, and hence a torque on the planet, which has been estimated using analytical methods. This torque, $\Gamma$, acts to change the planet's angular momentum, and hence its orbital radius, causing it to 'migrate', via: $$ \frac{\mathrm{d} L}{\mathrm{~d} t}=\Gamma $$ It is convenient to write the torque in terms of the reference value $$ \Gamma_{0}=\left(\frac{q}{h}\right)^{2} \Sigma_{p} r_{p}^{4} \Omega_{p}^{2} $$ c. From 2-dimensional steady fluid-dynamical disc models, it is predicted that the total torque $\Gamma$ has two main contributions: from the spiral wave, the 'Lindblad torque', $\Gamma_{L}$, and from the co-orbital region, the 'Corotation torque', $\Gamma_{C}$. For a disc of uniform entropy ( $\left.\mathrm{d} s=0\right)$, and with surface density profile $\Sigma \propto r^{-\alpha}$, and pressure profile $P \propto r^{-\delta}$, Tanaka et al. (2002) and Paardekooper \& Papaloizou (2009) find these torques are given by: $$ \begin{gathered} \Gamma_{L}=(-3.20+0.86 \alpha-2.33 \delta) \Gamma_{0} \\ \Gamma_{C}=5.97(1.5-\alpha) \Gamma_{0} \end{gathered} $$ We assume the gas in the disc obeys the ideal gas law, so that: $$ \frac{P}{\Sigma T}=\text { constant }, \quad \mathrm{d} s=\text { constant } \times\left(\frac{1}{\gamma-1} \frac{\mathrm{d} T}{T}-\frac{\mathrm{d} \Sigma}{\Sigma}\right), $$ where $T$ is the absolute temperature and $\gamma$ is the adiabatic index (the ratio of the heat capacity at constant pressure to the heat capacity at constant volume). Show that for a disc of uniform entropy, $$ \Gamma=\Gamma_{L}+\Gamma_{C}=(5.76-(5.11+2.33 \gamma) \alpha) \Gamma_{0} $$ [Hint: if $\frac{\mathrm{d} y}{y}=\lambda \frac{\mathrm{d} x}{x}$, then $y \propto x^{\lambda}$.] ## Helpful equations: The moment of inertia, $I$, of a point mass $m$ moving in a circle of radius $r$ is $I=m r^{2}$. The angular momentum, $L$, of a spinning object with an angular velocity of $\Omega$ is $L=I \Omega=r \times p$, where $p$ is the linear momentum of a point particle a distance $r$ from the axis of rotation.a. Consider a planet of mass $M_{p}$ in a circular orbit of radius $r_{p}$ about a star of mass $M_{*}$, with $M_{p} / M_{*}=q$ $<<1$. Show that it has angular velocity and angular momentum about the central star given by $\Omega_{\mathrm{p}}=\mathrm{V}\left(\mathrm{GM} * / \mathrm{r}_{\mathrm{p}}{ }^{3}\right)$, and $\mathrm{L}=\mathrm{M}_{\mathrm{p}} \mathrm{V}\left(\mathrm{GM} * \mathrm{r}_{\mathrm{p}}\right)$

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is an expression. Here is some context information for this question, which might assist you in solving it: Young, Earth-like planets interact with the protoplanetary discs in which they form, and as a result migrate to different orbital radii. The aim of this question is to quantify this migration in a simple model. We shall think of protoplanetary discs as consisting of nested circular orbits of gas and dust around a central star. For a thin disc (with small vertical extent), we assign to the disc a surface density (or mass per unit area) $\Sigma$, and semi-thickness $H$, which in general vary over the disc's extent. The disc's 'aspect ratio' at radius $r$ from the central star is denoted $h=H / r$. This question is concerned with the migration of 'small' planets, such that $M_{p} / M_{\star}=q \ll h^{3}$. [figure1] Figure 6: Left: ALMA image of the young star HL Tau and its protoplanetary disk. This image of planet formation reveals multiple rings and gaps that herald the presence of emerging planets as they sweep their orbits clear of dust and gas. Credit: ALMA (NRAO/ESO/NAOJ) / C. Brogan, B. Saxton (NRAO/AUI/NSF). Right: A small planet orbits whilst embedded in a protoplanetary disc, exciting a 1-armed spiral density wave. Credit: Frdric Masset. Since the planet is assumed small $\left(q \ll h^{3}\right)$, its interaction with the gas in the disc constitutes the excitation of a spiral density wave, and redistribution of matter in the co-orbital region (that is, matter orbiting at radii $r \approx r_{p}$ ), as shown in Figure 6. The resulting non-uniform density distribution induced in the disc exerts a net gravitational force, and hence a torque on the planet, which has been estimated using analytical methods. This torque, $\Gamma$, acts to change the planet's angular momentum, and hence its orbital radius, causing it to 'migrate', via: $$ \frac{\mathrm{d} L}{\mathrm{~d} t}=\Gamma $$ It is convenient to write the torque in terms of the reference value $$ \Gamma_{0}=\left(\frac{q}{h}\right)^{2} \Sigma_{p} r_{p}^{4} \Omega_{p}^{2} $$ c. From 2-dimensional steady fluid-dynamical disc models, it is predicted that the total torque $\Gamma$ has two main contributions: from the spiral wave, the 'Lindblad torque', $\Gamma_{L}$, and from the co-orbital region, the 'Corotation torque', $\Gamma_{C}$. For a disc of uniform entropy ( $\left.\mathrm{d} s=0\right)$, and with surface density profile $\Sigma \propto r^{-\alpha}$, and pressure profile $P \propto r^{-\delta}$, Tanaka et al. (2002) and Paardekooper \& Papaloizou (2009) find these torques are given by: $$ \begin{gathered} \Gamma_{L}=(-3.20+0.86 \alpha-2.33 \delta) \Gamma_{0} \\ \Gamma_{C}=5.97(1.5-\alpha) \Gamma_{0} \end{gathered} $$ We assume the gas in the disc obeys the ideal gas law, so that: $$ \frac{P}{\Sigma T}=\text { constant }, \quad \mathrm{d} s=\text { constant } \times\left(\frac{1}{\gamma-1} \frac{\mathrm{d} T}{T}-\frac{\mathrm{d} \Sigma}{\Sigma}\right), $$ where $T$ is the absolute temperature and $\gamma$ is the adiabatic index (the ratio of the heat capacity at constant pressure to the heat capacity at constant volume). Show that for a disc of uniform entropy, $$ \Gamma=\Gamma_{L}+\Gamma_{C}=(5.76-(5.11+2.33 \gamma) \alpha) \Gamma_{0} $$ [Hint: if $\frac{\mathrm{d} y}{y}=\lambda \frac{\mathrm{d} x}{x}$, then $y \propto x^{\lambda}$.] ## Helpful equations: The moment of inertia, $I$, of a point mass $m$ moving in a circle of radius $r$ is $I=m r^{2}$. The angular momentum, $L$, of a spinning object with an angular velocity of $\Omega$ is $L=I \Omega=r \times p$, where $p$ is the linear momentum of a point particle a distance $r$ from the axis of rotation. problem: a. Consider a planet of mass $M_{p}$ in a circular orbit of radius $r_{p}$ about a star of mass $M_{*}$, with $M_{p} / M_{*}=q$ $<<1$. Show that it has angular velocity and angular momentum about the central star given by $\Omega_{\mathrm{p}}=\mathrm{V}\left(\mathrm{GM} * / \mathrm{r}_{\mathrm{p}}{ }^{3}\right)$, and $\mathrm{L}=\mathrm{M}_{\mathrm{p}} \mathrm{V}\left(\mathrm{GM} * \mathrm{r}_{\mathrm{p}}\right)$ All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is an expression without equals signs, e.g. ANSWER=\frac{1}{2} g t^2

EX

Astronomy

EN

multi-modal

Astronomy_540

如图所示, $a$ 为放在赤道上相对地球静止的物体随地球自转做匀速圆周运动, $b$ 为在地球表面附近做匀速圆周运动的人造卫星 (轨道半径近似等于地球半径), $c$ 为地球的同步卫星, 则以下说法中正确的是（ ）[图1] A: $a 、 b 、 c$ 的向心加速度大小关系为 $\mathrm{a}_{\mathrm{b}}>\mathrm{a}_{\mathrm{c}}>\mathrm{a}_{\mathrm{a}}$ B: $a 、 b 、 c$ 的角速度大小关系为 $\omega_{\mathrm{a}}>\omega_{\mathrm{b}}>\omega_{\mathrm{c}}$ C: $a 、 b 、 c$ 的线速度大小关系为 $\mathrm{v}_{\mathrm{a}}<\mathrm{v}_{\mathrm{b}}<\mathrm{v}_{\mathrm{c}}$ D: $a 、 b 、 c$ 的周期大小关系为 $\mathrm{T}_{\mathrm{a}}<\mathrm{T}_{\mathrm{b}}<\mathrm{T}_{\mathrm{c}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 如图所示, $a$ 为放在赤道上相对地球静止的物体随地球自转做匀速圆周运动, $b$ 为在地球表面附近做匀速圆周运动的人造卫星 (轨道半径近似等于地球半径), $c$ 为地球的同步卫星, 则以下说法中正确的是（ ）[图1] A: $a 、 b 、 c$ 的向心加速度大小关系为 $\mathrm{a}_{\mathrm{b}}>\mathrm{a}_{\mathrm{c}}>\mathrm{a}_{\mathrm{a}}$ B: $a 、 b 、 c$ 的角速度大小关系为 $\omega_{\mathrm{a}}>\omega_{\mathrm{b}}>\omega_{\mathrm{c}}$ C: $a 、 b 、 c$ 的线速度大小关系为 $\mathrm{v}_{\mathrm{a}}<\mathrm{v}_{\mathrm{b}}<\mathrm{v}_{\mathrm{c}}$ D: $a 、 b 、 c$ 的周期大小关系为 $\mathrm{T}_{\mathrm{a}}<\mathrm{T}_{\mathrm{b}}<\mathrm{T}_{\mathrm{c}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

multi-modal

Astronomy_453

2018 年 6 月 14 日 11 时 06 分，探月工程嫦娥四号任务“鹊桥”中继星成为世界首颗成功进入地月拉格朗日 $L_{2}$ 点的 Halo 使命轨道的卫星, 为地月信息联通搭建“天桥”。如图所示, 该 $L_{2}$ 点位于地球与月球连线的延长线上, “鹊桥”位于该点, 在几乎不消耗燃料的情况下与月球同步绕地球做圆周运动。已知地球、月球和“鹊桥”的质量分别为 $M_{e}$ 、 $M_{m} 、 m$ ，地球和月球之间的平均距离为 $R, L_{2}$ 点离月球的距离为 $x$, 则 [图1] A: “鹊桥”的线速度大于月球的线速度 B: “鹊桥”的向心加速度小于月球的向心加速度 C: $x$ 满足 $\frac{M_{e}}{(R+x)^{2}}+\frac{M_{m}}{x^{2}}=\frac{M_{e}}{R^{3}}(R+x)$ D: $x$ 满足 $\frac{M_{e}}{(R+x)^{2}}+\frac{M_{e}}{x^{2}}=\frac{m}{R^{3}}(R+x)$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 2018 年 6 月 14 日 11 时 06 分，探月工程嫦娥四号任务“鹊桥”中继星成为世界首颗成功进入地月拉格朗日 $L_{2}$ 点的 Halo 使命轨道的卫星, 为地月信息联通搭建“天桥”。如图所示, 该 $L_{2}$ 点位于地球与月球连线的延长线上, “鹊桥”位于该点, 在几乎不消耗燃料的情况下与月球同步绕地球做圆周运动。已知地球、月球和“鹊桥”的质量分别为 $M_{e}$ 、 $M_{m} 、 m$ ，地球和月球之间的平均距离为 $R, L_{2}$ 点离月球的距离为 $x$, 则 [图1] A: “鹊桥”的线速度大于月球的线速度 B: “鹊桥”的向心加速度小于月球的向心加速度 C: $x$ 满足 $\frac{M_{e}}{(R+x)^{2}}+\frac{M_{m}}{x^{2}}=\frac{M_{e}}{R^{3}}(R+x)$ D: $x$ 满足 $\frac{M_{e}}{(R+x)^{2}}+\frac{M_{e}}{x^{2}}=\frac{m}{R^{3}}(R+x)$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_1001

Although Saturn is famous for its rings, all of the gas giants in the Solar System have ring systems. The outer ring is known as the Adams ring and is very thin. Normally such a thin structure would widen over time so there needs to be a process keeping it constrained. One hypothesis is that the Neptunian moon Galatea, with an orbit just slightly smaller than the ring, acts as a 'shepherd moon' by having a 42 : 43 orbital resonance with particles in the ring, in terms of the period of their orbits. The ring and the moon are shown in Figure 1. [figure1] Figure 1: Left: Neptune as seen by the Voyager 2 mission in August 1989, a few days before its flyby. Credit: NASA / JPL / Voyager-ISS / Justin Cowart. Right: Neptune and its ring system as imaged in the infrared by the NIRCam instrument on the James Webb Space Telescope in July 2022. Multiple moons and rings are visible, with Galatea and the Adams ring labelled. Credit: NASA / ESA / CSA / STScI / Joseph DePasquale. The semi-major axis of Galatea is $61953 \mathrm{~km}$. Assume the moon and the ring particles travel in circular orbits. Measurements of the amplitude of ripples in the ring caused by gravitational attraction towards the moon indicate the ring particles closest to the moon have a radial acceleration of $0.15 \mathrm{~mm} \mathrm{~s}^{-1}$. Estimate the mass of the moon.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. problem: Although Saturn is famous for its rings, all of the gas giants in the Solar System have ring systems. The outer ring is known as the Adams ring and is very thin. Normally such a thin structure would widen over time so there needs to be a process keeping it constrained. One hypothesis is that the Neptunian moon Galatea, with an orbit just slightly smaller than the ring, acts as a 'shepherd moon' by having a 42 : 43 orbital resonance with particles in the ring, in terms of the period of their orbits. The ring and the moon are shown in Figure 1. [figure1] Figure 1: Left: Neptune as seen by the Voyager 2 mission in August 1989, a few days before its flyby. Credit: NASA / JPL / Voyager-ISS / Justin Cowart. Right: Neptune and its ring system as imaged in the infrared by the NIRCam instrument on the James Webb Space Telescope in July 2022. Multiple moons and rings are visible, with Galatea and the Adams ring labelled. Credit: NASA / ESA / CSA / STScI / Joseph DePasquale. The semi-major axis of Galatea is $61953 \mathrm{~km}$. Assume the moon and the ring particles travel in circular orbits. Measurements of the amplitude of ripples in the ring caused by gravitational attraction towards the moon indicate the ring particles closest to the moon have a radial acceleration of $0.15 \mathrm{~mm} \mathrm{~s}^{-1}$. Estimate the mass of the moon. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of kg, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

multi-modal

Astronomy_663

运城市位于山西省南端黄河金三角地区，与陕西、河南两省隔黄河而相望，近年来随着经济的增长, 环境却越来越糟糕: 冬天的雾霔极为严重, 某些企业对水和空气的污染让周边的居民苦不堪言...... 为对运城市及相邻区域内的环境进行有效的监测, 现计划发射一颗环境监测卫星, 要求该卫星每天同一时刻以相同的方向通过运城市的正上空,已知运城市的地理坐标在东经 $110^{\circ} 30^{\prime}$ 和北纬 $35^{\circ} 30^{\prime}$, 下列四选项为康杰中学的同学们对该卫星的运行参数的讨论稿，其中说法正确的是（ ） A: 该卫星轨道可以是周期为 120 分钟的近极地轨道 B: 该卫星的轨道必须为地球同步卫星轨道 C: 该卫星的轨道平面必过地心且与赤道平面呈 $35^{\circ} 30^{\prime}$ 的夹角 D: 该卫星的转动角速度约为 $7.3 \times 10^{-5} \mathrm{rad} / \mathrm{s}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 运城市位于山西省南端黄河金三角地区，与陕西、河南两省隔黄河而相望，近年来随着经济的增长, 环境却越来越糟糕: 冬天的雾霔极为严重, 某些企业对水和空气的污染让周边的居民苦不堪言...... 为对运城市及相邻区域内的环境进行有效的监测, 现计划发射一颗环境监测卫星, 要求该卫星每天同一时刻以相同的方向通过运城市的正上空,已知运城市的地理坐标在东经 $110^{\circ} 30^{\prime}$ 和北纬 $35^{\circ} 30^{\prime}$, 下列四选项为康杰中学的同学们对该卫星的运行参数的讨论稿，其中说法正确的是（ ） A: 该卫星轨道可以是周期为 120 分钟的近极地轨道 B: 该卫星的轨道必须为地球同步卫星轨道 C: 该卫星的轨道平面必过地心且与赤道平面呈 $35^{\circ} 30^{\prime}$ 的夹角 D: 该卫星的转动角速度约为 $7.3 \times 10^{-5} \mathrm{rad} / \mathrm{s}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

text-only

Astronomy_1223

GW170817 was the first gravitational wave event arising from a binary neutron star merger to have been detected by the LIGO \& Virgo experiments, and careful localization of the source meant that the electromagnetic counterpart was quickly found in galaxy NGC 4993 (see Figure 2). Such a combination of two completely separate branches of astronomical observation begins a new era of 'multi-messenger astronomy'. Since the gravitational waves allow an independent measurement of the distance to the host galaxy and the light allows an independent measurement of the recessional speed, this observation allows us to determine a new, independent value of the Hubble constant $H_{0}$. Another way of measuring distances to galaxies is to use the Fundamental Plane (FP) relation, which relies on the assumption there is a fairly tight relation between radius, surface brightness, and velocity dispersion for bulge-dominated galaxies, and is widely used for galaxies like NGC 4993. It can be described by the relation $$ \log \left(\frac{D}{(1+z)^{2}}\right)=-\log R_{e}+\alpha \log \sigma-\beta \log \left\langle I_{r}\right\rangle_{e}+\gamma $$ where $D$ is the distance in Mpc, $R_{e}$ is the effective radius measured in arcseconds, $\sigma$ is the velocity dispersion in $\mathrm{km} \mathrm{s}^{-1},\left\langle I_{r}\right\rangle_{e}$ is the mean intensity inside the effective radius measured in $L_{\odot} \mathrm{pc}^{-2}$, and $\gamma$ is the distance-dependent zero point of the relation. Calibrating the zero point to the Leo I galaxy group, the constants in the FP relation become $\alpha=1.24, \beta=0.82$, and $\gamma=2.194$. [figure1] Figure 2: Left: The GW170817 signal as measured by the LIGO and Virgo gravitational wave detectors, taken from Abbott $e t$ al. (2017). The normalized amplitude (or strain) is in units of $10^{-21}$. The signal is not visible in the Virgo data due to the direction of the source with respect to the detector's antenna pattern. Right: The optical counterpart of GW170817 in host galaxy NGC 4993, taken from Hjorth et al. (2017). By measuring the amplitude (called strain, $h$ ) and the frequency of the gravitational waves, $f_{\mathrm{GW}}$, one can determine the distance to the source without having to rely on 'standard candles' like Cepheid variables or Type Ia supernovae. For two masses, $m_{1}$ and $m_{2}$, orbiting the centre of mass with separation $a$ with orbital angular velocity $\omega$, then the dimensionless strain parameter $h$ is $$ h \simeq \frac{G}{c^{4}} \frac{1}{r} \mu a^{2} \omega^{2} $$ where $r$ is the luminosity distance, $c$ is the speed of light, $\mu=m_{1} m_{2} / M_{\text {tot }}$ is the reduced mass and $M_{\text {tot }}=m_{1}+m_{2}$ is the total mass.c. The Wolf-Rayet star WR7 is in the constellation of Canis Major and its strong winds are responsible for the nebula known as Thor's Helmet. The star has a mass of $16 M_{\odot}$, a radius of $1.41 R_{\odot}$ and a surface temperature of $112000 \mathrm{~K}$, with a measured $v_{\infty}$ of $1545 \mathrm{~km} \mathrm{~s}^{-1}$. ii. The nebula is 5 arcmins in diameter $\left(1^{\circ}=60 \mathrm{arcmin}\right)$ and $4.8 \mathrm{kpc}$ away, and at its edge is a bright thin shell of swept up material expanding at a rate of $30 \mathrm{~km} \mathrm{~s}^{-1}$. The age of such a nebula, $t$, is related to the current values of radius, $R$, and expansion speed, $v$, by $t=0.55 R / v$. Using this model, determine the age of the nebula.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: GW170817 was the first gravitational wave event arising from a binary neutron star merger to have been detected by the LIGO \& Virgo experiments, and careful localization of the source meant that the electromagnetic counterpart was quickly found in galaxy NGC 4993 (see Figure 2). Such a combination of two completely separate branches of astronomical observation begins a new era of 'multi-messenger astronomy'. Since the gravitational waves allow an independent measurement of the distance to the host galaxy and the light allows an independent measurement of the recessional speed, this observation allows us to determine a new, independent value of the Hubble constant $H_{0}$. Another way of measuring distances to galaxies is to use the Fundamental Plane (FP) relation, which relies on the assumption there is a fairly tight relation between radius, surface brightness, and velocity dispersion for bulge-dominated galaxies, and is widely used for galaxies like NGC 4993. It can be described by the relation $$ \log \left(\frac{D}{(1+z)^{2}}\right)=-\log R_{e}+\alpha \log \sigma-\beta \log \left\langle I_{r}\right\rangle_{e}+\gamma $$ where $D$ is the distance in Mpc, $R_{e}$ is the effective radius measured in arcseconds, $\sigma$ is the velocity dispersion in $\mathrm{km} \mathrm{s}^{-1},\left\langle I_{r}\right\rangle_{e}$ is the mean intensity inside the effective radius measured in $L_{\odot} \mathrm{pc}^{-2}$, and $\gamma$ is the distance-dependent zero point of the relation. Calibrating the zero point to the Leo I galaxy group, the constants in the FP relation become $\alpha=1.24, \beta=0.82$, and $\gamma=2.194$. [figure1] Figure 2: Left: The GW170817 signal as measured by the LIGO and Virgo gravitational wave detectors, taken from Abbott $e t$ al. (2017). The normalized amplitude (or strain) is in units of $10^{-21}$. The signal is not visible in the Virgo data due to the direction of the source with respect to the detector's antenna pattern. Right: The optical counterpart of GW170817 in host galaxy NGC 4993, taken from Hjorth et al. (2017). By measuring the amplitude (called strain, $h$ ) and the frequency of the gravitational waves, $f_{\mathrm{GW}}$, one can determine the distance to the source without having to rely on 'standard candles' like Cepheid variables or Type Ia supernovae. For two masses, $m_{1}$ and $m_{2}$, orbiting the centre of mass with separation $a$ with orbital angular velocity $\omega$, then the dimensionless strain parameter $h$ is $$ h \simeq \frac{G}{c^{4}} \frac{1}{r} \mu a^{2} \omega^{2} $$ where $r$ is the luminosity distance, $c$ is the speed of light, $\mu=m_{1} m_{2} / M_{\text {tot }}$ is the reduced mass and $M_{\text {tot }}=m_{1}+m_{2}$ is the total mass. problem: c. The Wolf-Rayet star WR7 is in the constellation of Canis Major and its strong winds are responsible for the nebula known as Thor's Helmet. The star has a mass of $16 M_{\odot}$, a radius of $1.41 R_{\odot}$ and a surface temperature of $112000 \mathrm{~K}$, with a measured $v_{\infty}$ of $1545 \mathrm{~km} \mathrm{~s}^{-1}$. ii. The nebula is 5 arcmins in diameter $\left(1^{\circ}=60 \mathrm{arcmin}\right)$ and $4.8 \mathrm{kpc}$ away, and at its edge is a bright thin shell of swept up material expanding at a rate of $30 \mathrm{~km} \mathrm{~s}^{-1}$. The age of such a nebula, $t$, is related to the current values of radius, $R$, and expansion speed, $v$, by $t=0.55 R / v$. Using this model, determine the age of the nebula. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of s, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

multi-modal

Astronomy_872

Two protons $A$ and $B$ lie in the solar interior. In the rest frame of proton $A$, the proton $B$ approaches it radially from a large distance with speed $0.9 c$. In the rest frame of proton A, identify the radius of the "classically forbidden" region for proton B (i.e. the region in which proton B cannot enter). A: $6.6 \times 10^{-12} \mathrm{~m}$ B: $4.1 \times 10^{-15} \mathrm{~m}$ C: $2.3 \times 10^{-15} \mathrm{~m}$ D: $3.8 \times 10^{-18} \mathrm{~m}$ E: $1.2 \times 10^{-18} \mathrm{~m}$

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: Two protons $A$ and $B$ lie in the solar interior. In the rest frame of proton $A$, the proton $B$ approaches it radially from a large distance with speed $0.9 c$. In the rest frame of proton A, identify the radius of the "classically forbidden" region for proton B (i.e. the region in which proton B cannot enter). A: $6.6 \times 10^{-12} \mathrm{~m}$ B: $4.1 \times 10^{-15} \mathrm{~m}$ C: $2.3 \times 10^{-15} \mathrm{~m}$ D: $3.8 \times 10^{-18} \mathrm{~m}$ E: $1.2 \times 10^{-18} \mathrm{~m}$ You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D, E].

SC

Astronomy

EN

text-only

Astronomy_805

For the five maneuvers described above, rank the resulting apogees from lowest to highest. Assume the change in velocity is small relative to orbital velocity, but not negligible. A: $2<3=4=5<1$ B: $2=3<5<4=1$ C: $2<3=4<5<1$ D: $2<5<3=4<1$ E: $2<3<5<4<1$

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: For the five maneuvers described above, rank the resulting apogees from lowest to highest. Assume the change in velocity is small relative to orbital velocity, but not negligible. A: $2<3=4=5<1$ B: $2=3<5<4=1$ C: $2<3=4<5<1$ D: $2<5<3=4<1$ E: $2<3<5<4<1$ You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D, E].

SC

Astronomy

EN

text-only

Astronomy_758

The Jovian planets are ... A: Mercury, Venus, Earth, Mars B: Mercury, Venus, Earth, Mars, Jupiter, Saturn C: Jupiter, Saturn, Uranus, Neptune D: Uranus, Neptune

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: The Jovian planets are ... A: Mercury, Venus, Earth, Mars B: Mercury, Venus, Earth, Mars, Jupiter, Saturn C: Jupiter, Saturn, Uranus, Neptune D: Uranus, Neptune You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

SC

Astronomy

EN

text-only

Astronomy_877

A comet's orbit has the following characteristics: eccentricity $\mathrm{e}=0.995$; aphelion distance $r_{a}=5 \cdot 10^{4} \mathrm{AU}$. Assume we know the mass of the Sun $M_{S}=1.98 \cdot 10^{30} \mathrm{~kg}$, and gravitational constant $G=6.67 \cdot 10^{-11} \mathrm{Nm}^{2} / \mathrm{kg}^{2}$. Determine the velocity of the comet at its aphelion. A: $34.76 \mathrm{~m} / \mathrm{s}$ B: $20.57 \mathrm{~m} / \mathrm{s}$ C: $187.91 \mathrm{~m} / \mathrm{s}$ D: $63.38 \mathrm{~m} / \mathrm{s}$ E: $9.19 \mathrm{~m} / \mathrm{s}$

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: A comet's orbit has the following characteristics: eccentricity $\mathrm{e}=0.995$; aphelion distance $r_{a}=5 \cdot 10^{4} \mathrm{AU}$. Assume we know the mass of the Sun $M_{S}=1.98 \cdot 10^{30} \mathrm{~kg}$, and gravitational constant $G=6.67 \cdot 10^{-11} \mathrm{Nm}^{2} / \mathrm{kg}^{2}$. Determine the velocity of the comet at its aphelion. A: $34.76 \mathrm{~m} / \mathrm{s}$ B: $20.57 \mathrm{~m} / \mathrm{s}$ C: $187.91 \mathrm{~m} / \mathrm{s}$ D: $63.38 \mathrm{~m} / \mathrm{s}$ E: $9.19 \mathrm{~m} / \mathrm{s}$ You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D, E].

SC

Astronomy

EN

text-only

Astronomy_1184

On Earth the Global Positioning System (GPS) requires a minimum of 24 satellites in orbit at any one time (there are typically more than that to allow for redundancies, with the current constellation having more than 30) so that at least 4 are visible above the horizon from anywhere on Earth (necessary for an $\mathrm{x}, \mathrm{y}, \mathrm{z}$ and time co-ordinate). This is achieved by having 6 different orbital planes, separated by $60^{\circ}$, and each orbital plane has 4 satellites. [figure1] Figure 1: The current set up of the GPS system used on Earth. Credits: Left: Peter H. Dana, University of Colorado; Right: GPS Standard Positioning Service Specification, $4^{\text {th }}$ edition The orbits are essentially circular with an eccentricity $<0.02$, an inclination of $55^{\circ}$, and an orbital period of exactly half a sidereal day (called a semi-synchronous orbit). The receiving angle of each satellite's antenna needs to be about $27.8^{\circ}$, and hence about $38 \%$ of the Earth's surface is within each satellite's footprint (see Figure 1), allowing the excellent coverage required.c. Using suitable calculations, explore the viability of a 24-satellite GPS constellation similar to the one used on Earth, in a semi-synchronous Martian orbit, by considering: i. Would the moons prevent such an orbit?

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: On Earth the Global Positioning System (GPS) requires a minimum of 24 satellites in orbit at any one time (there are typically more than that to allow for redundancies, with the current constellation having more than 30) so that at least 4 are visible above the horizon from anywhere on Earth (necessary for an $\mathrm{x}, \mathrm{y}, \mathrm{z}$ and time co-ordinate). This is achieved by having 6 different orbital planes, separated by $60^{\circ}$, and each orbital plane has 4 satellites. [figure1] Figure 1: The current set up of the GPS system used on Earth. Credits: Left: Peter H. Dana, University of Colorado; Right: GPS Standard Positioning Service Specification, $4^{\text {th }}$ edition The orbits are essentially circular with an eccentricity $<0.02$, an inclination of $55^{\circ}$, and an orbital period of exactly half a sidereal day (called a semi-synchronous orbit). The receiving angle of each satellite's antenna needs to be about $27.8^{\circ}$, and hence about $38 \%$ of the Earth's surface is within each satellite's footprint (see Figure 1), allowing the excellent coverage required. problem: c. Using suitable calculations, explore the viability of a 24-satellite GPS constellation similar to the one used on Earth, in a semi-synchronous Martian orbit, by considering: i. Would the moons prevent such an orbit? All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of \mathrm{~km}$, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

multi-modal

Astronomy_1129

The Parker Solar Probe (PSP) is part of a mission to learn more about the Sun, named after the scientist that first proposed the existence of the solar wind, and was launched on $12^{\text {th }}$ August 2018. Over the course of the 7 year mission it will orbit the Sun 24 times, and through 7 flybys of Venus it will lose some energy in order to get into an ever tighter orbit (see Figure 1). In its final 3 orbits it will have a perihelion (closest approach to the Sun) of only $r_{\text {peri }}=9.86 R_{\odot}$, about 7 times closer than any previous probe, the first of which is due on $24^{\text {th }}$ December 2024. In this extreme environment the probe will not only face extreme brightness and temperatures but also will break the record for the fastest ever spacecraft. [figure1] Figure 1: Left: The journey PSP will take to get from the Earth to the final orbit around the Sun. Right: The probe just after assembly in the John Hopkins University Applied Physics Laboratory. Credit: NASA / John Hopkins APL / Ed Whitman. $$ v^{2}=G M\left(\frac{2}{r}-\frac{1}{a}\right) $$ Given that in its final orbit PSP has a orbital period of 88 days, calculate the speed of the probe as it passes through the minimum perihelion. Give your answer in $\mathrm{km} \mathrm{s}^{-1}$. Close to the Sun the communications equipment is very sensitive to the extreme environment, so the mission is planned for the probe to take all of its primary science measurements whilst within 0.25 au of the Sun, and then to spend the rest of the orbit beaming that data back to Earth, as shown in Figure 2. [figure2] Figure 2: The way PSP is planned to split each orbit into taking measurements and sending data back. Credit: NASA / Johns Hopkins APL. When considering the position of an object in an elliptical orbit as a function of time, there are two important angles (called 'anomalies') necessary to do the calculation, and they are defined in Figure 3. By constructing a circular orbit centred on the same point as the ellipse and with the same orbital period, the eccentric anomaly, $E$, is then the angle between the major axis and the perpendicular projection of the object (some time $t$ after perihelion) onto the circle as measured from the centre of the ellipse ( $\angle x c z$ in the figure). The mean anomaly, $M$, is the angle between the major axis and where the object would have been at time $t$ if it was indeed on the circular orbit ( $\angle y c z$ in the figure, such that the shaded areas are the same). [figure3] Figure 3: The definitions of the anomalies needed to get the position of an object in an ellipse as a function of time. The Sun (located at the focus) is labeled $S$ and the probe $P . M$ and $E$ are the mean and eccentric anomalies respectively. The angle $\theta$ is called the true anomaly and is not needed for this question. Credit: Wikipedia. The eccentric anomaly can be related to the mean anomaly through Kepler's Equation, $$ M=E-e \sin E \text {. } $$a. When the probe is at its closest perihelion: i. Calculate the apparent magnitude of the Sun, given that from Earth $m_{\odot}=-26.74$.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: The Parker Solar Probe (PSP) is part of a mission to learn more about the Sun, named after the scientist that first proposed the existence of the solar wind, and was launched on $12^{\text {th }}$ August 2018. Over the course of the 7 year mission it will orbit the Sun 24 times, and through 7 flybys of Venus it will lose some energy in order to get into an ever tighter orbit (see Figure 1). In its final 3 orbits it will have a perihelion (closest approach to the Sun) of only $r_{\text {peri }}=9.86 R_{\odot}$, about 7 times closer than any previous probe, the first of which is due on $24^{\text {th }}$ December 2024. In this extreme environment the probe will not only face extreme brightness and temperatures but also will break the record for the fastest ever spacecraft. [figure1] Figure 1: Left: The journey PSP will take to get from the Earth to the final orbit around the Sun. Right: The probe just after assembly in the John Hopkins University Applied Physics Laboratory. Credit: NASA / John Hopkins APL / Ed Whitman. $$ v^{2}=G M\left(\frac{2}{r}-\frac{1}{a}\right) $$ Given that in its final orbit PSP has a orbital period of 88 days, calculate the speed of the probe as it passes through the minimum perihelion. Give your answer in $\mathrm{km} \mathrm{s}^{-1}$. Close to the Sun the communications equipment is very sensitive to the extreme environment, so the mission is planned for the probe to take all of its primary science measurements whilst within 0.25 au of the Sun, and then to spend the rest of the orbit beaming that data back to Earth, as shown in Figure 2. [figure2] Figure 2: The way PSP is planned to split each orbit into taking measurements and sending data back. Credit: NASA / Johns Hopkins APL. When considering the position of an object in an elliptical orbit as a function of time, there are two important angles (called 'anomalies') necessary to do the calculation, and they are defined in Figure 3. By constructing a circular orbit centred on the same point as the ellipse and with the same orbital period, the eccentric anomaly, $E$, is then the angle between the major axis and the perpendicular projection of the object (some time $t$ after perihelion) onto the circle as measured from the centre of the ellipse ( $\angle x c z$ in the figure). The mean anomaly, $M$, is the angle between the major axis and where the object would have been at time $t$ if it was indeed on the circular orbit ( $\angle y c z$ in the figure, such that the shaded areas are the same). [figure3] Figure 3: The definitions of the anomalies needed to get the position of an object in an ellipse as a function of time. The Sun (located at the focus) is labeled $S$ and the probe $P . M$ and $E$ are the mean and eccentric anomalies respectively. The angle $\theta$ is called the true anomaly and is not needed for this question. Credit: Wikipedia. The eccentric anomaly can be related to the mean anomaly through Kepler's Equation, $$ M=E-e \sin E \text {. } $$ problem: a. When the probe is at its closest perihelion: i. Calculate the apparent magnitude of the Sun, given that from Earth $m_{\odot}=-26.74$. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value.

NV

Astronomy

EN

multi-modal

Astronomy_1193

On Earth the Global Positioning System (GPS) requires a minimum of 24 satellites in orbit at any one time (there are typically more than that to allow for redundancies, with the current constellation having more than 30) so that at least 4 are visible above the horizon from anywhere on Earth (necessary for an $\mathrm{x}, \mathrm{y}, \mathrm{z}$ and time co-ordinate). This is achieved by having 6 different orbital planes, separated by $60^{\circ}$, and each orbital plane has 4 satellites. [figure1] Figure 1: The current set up of the GPS system used on Earth. Credits: Left: Peter H. Dana, University of Colorado; Right: GPS Standard Positioning Service Specification, $4^{\text {th }}$ edition The orbits are essentially circular with an eccentricity $<0.02$, an inclination of $55^{\circ}$, and an orbital period of exactly half a sidereal day (called a semi-synchronous orbit). The receiving angle of each satellite's antenna needs to be about $27.8^{\circ}$, and hence about $38 \%$ of the Earth's surface is within each satellite's footprint (see Figure 1), allowing the excellent coverage required.ii. How would the GPS positional accuracy compare to Earth?

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: On Earth the Global Positioning System (GPS) requires a minimum of 24 satellites in orbit at any one time (there are typically more than that to allow for redundancies, with the current constellation having more than 30) so that at least 4 are visible above the horizon from anywhere on Earth (necessary for an $\mathrm{x}, \mathrm{y}, \mathrm{z}$ and time co-ordinate). This is achieved by having 6 different orbital planes, separated by $60^{\circ}$, and each orbital plane has 4 satellites. [figure1] Figure 1: The current set up of the GPS system used on Earth. Credits: Left: Peter H. Dana, University of Colorado; Right: GPS Standard Positioning Service Specification, $4^{\text {th }}$ edition The orbits are essentially circular with an eccentricity $<0.02$, an inclination of $55^{\circ}$, and an orbital period of exactly half a sidereal day (called a semi-synchronous orbit). The receiving angle of each satellite's antenna needs to be about $27.8^{\circ}$, and hence about $38 \%$ of the Earth's surface is within each satellite's footprint (see Figure 1), allowing the excellent coverage required. problem: ii. How would the GPS positional accuracy compare to Earth? All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of \mathrm{~m}, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

multi-modal

Astronomy_303

新时代的中国北斗导航系统是世界一流的。空间段由若干地球静止轨道卫星、倾斜地球同步轨道卫星和中圆地球轨道卫星组成。已知地球表面两极处的重力加速度为 $g_{0}$,赤道处的重力加速度为 $g_{1}$, 万有引力常量为 $G$ 。若把地球看成密度均匀、半径为 $R$ 的球 体，下列说法正确的是（） A: 北斗地球同步卫星距离地球表面的高度 $h=\left(\sqrt[3]{\frac{g_{1}}{g_{0}-g_{1}}}-1\right) R$ B: 北斗地球同步卫星距离地球表面的高度 $h=\left(\sqrt[3]{\frac{g_{0}}{g_{0}-g_{1}}}-1\right) R$ C: 地球的平均密度 $\rho=\frac{3 g_{1}}{4 \pi G R}$ D: 地球的近地卫星的周期 $T_{0}=2 \pi \sqrt{\frac{R}{g_{1}}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 新时代的中国北斗导航系统是世界一流的。空间段由若干地球静止轨道卫星、倾斜地球同步轨道卫星和中圆地球轨道卫星组成。已知地球表面两极处的重力加速度为 $g_{0}$,赤道处的重力加速度为 $g_{1}$, 万有引力常量为 $G$ 。若把地球看成密度均匀、半径为 $R$ 的球 体，下列说法正确的是（） A: 北斗地球同步卫星距离地球表面的高度 $h=\left(\sqrt[3]{\frac{g_{1}}{g_{0}-g_{1}}}-1\right) R$ B: 北斗地球同步卫星距离地球表面的高度 $h=\left(\sqrt[3]{\frac{g_{0}}{g_{0}-g_{1}}}-1\right) R$ C: 地球的平均密度 $\rho=\frac{3 g_{1}}{4 \pi G R}$ D: 地球的近地卫星的周期 $T_{0}=2 \pi \sqrt{\frac{R}{g_{1}}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy_966

"Manhattanhenge" is the name given to when, just before sunset or just after sunrise 4 times a year (twice for setting, and twice for rising), the Sun aligns with the east-west streets of the New York grid system. One of the setting dates this year was on 11th July. Which of these is another date you are likely to see the "Manhattanhenge" sunset? A: $30^{\text {th }}$ May B: $30^{\text {th }}$ June C: $11^{\text {th }}$ December D: $11^{\text {th }}$ January

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: "Manhattanhenge" is the name given to when, just before sunset or just after sunrise 4 times a year (twice for setting, and twice for rising), the Sun aligns with the east-west streets of the New York grid system. One of the setting dates this year was on 11th July. Which of these is another date you are likely to see the "Manhattanhenge" sunset? A: $30^{\text {th }}$ May B: $30^{\text {th }}$ June C: $11^{\text {th }}$ December D: $11^{\text {th }}$ January You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

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Astronomy_247

2022 年 2 月 27 日上午, 长征八号遥二运载火箭在文昌航天发射场点火升空, 成功将 22 颗卫星送入预定轨道，这次发射，创造了我国一箭多星发射的新纪录。已知其中一颗卫星绕地球运行近似为匀速圆周运动, 到地面距离为 $h$, 地球半径为 $R$, 地球表面的重力加速度为 $g$, 下列说法正确的是 ( ) A: 该卫星的向心加速度小于 $g$ B: 该卫星的运行速度有可能等于第一宇宙速度 C: 由题干条件无法求出地球的质量 D: 由于稀薄大气的阻力影响, 该卫星运行的轨道半径会变小, 速度也变小

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 2022 年 2 月 27 日上午, 长征八号遥二运载火箭在文昌航天发射场点火升空, 成功将 22 颗卫星送入预定轨道，这次发射，创造了我国一箭多星发射的新纪录。已知其中一颗卫星绕地球运行近似为匀速圆周运动, 到地面距离为 $h$, 地球半径为 $R$, 地球表面的重力加速度为 $g$, 下列说法正确的是 ( ) A: 该卫星的向心加速度小于 $g$ B: 该卫星的运行速度有可能等于第一宇宙速度 C: 由题干条件无法求出地球的质量 D: 由于稀薄大气的阻力影响, 该卫星运行的轨道半径会变小, 速度也变小 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

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Astronomy_721

地球赤道上有一物体随地球的自转而做圆周运动,所受的向心力为 $F_{l}$, 向心加速度为 $a_{1}$, 线速度为 $v_{1}$, 角速度为 $\omega_{1}$; 绕地球表面附近做圆周运动的人造卫星受的向心力为 $F_{2}$, 向心加速度为 $a_{2}$, 线速度为 $v_{2}$, 角速度为 $\omega_{2}$; 地球同步卫星所受的向心力为 $F_{3}$, 向心加速度为 $a_{3}$, 线速度为 $v_{3}$, 角速度为 $\omega_{3}$. 地球表面重力加速度为 $g$,第一宇宙速度为 $\mathrm{v}$,假设三者质量相等,下列结论中错误的是( ) A: $F_{2}>F_{3}>F_{1}$ B: $a_{2}=g>a_{3}>a_{1}$ C: $v_{1}=v_{2}=v>v_{3}$ D: $\omega_{1}=\omega_{3}<\omega_{2}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 地球赤道上有一物体随地球的自转而做圆周运动,所受的向心力为 $F_{l}$, 向心加速度为 $a_{1}$, 线速度为 $v_{1}$, 角速度为 $\omega_{1}$; 绕地球表面附近做圆周运动的人造卫星受的向心力为 $F_{2}$, 向心加速度为 $a_{2}$, 线速度为 $v_{2}$, 角速度为 $\omega_{2}$; 地球同步卫星所受的向心力为 $F_{3}$, 向心加速度为 $a_{3}$, 线速度为 $v_{3}$, 角速度为 $\omega_{3}$. 地球表面重力加速度为 $g$,第一宇宙速度为 $\mathrm{v}$,假设三者质量相等,下列结论中错误的是( ) A: $F_{2}>F_{3}>F_{1}$ B: $a_{2}=g>a_{3}>a_{1}$ C: $v_{1}=v_{2}=v>v_{3}$ D: $\omega_{1}=\omega_{3}<\omega_{2}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy_308

某天体科学家在太阳系外发现一颗类地球行星, 这颗类地行星绕中心恒星做圆周运动, 公转的周期为 146 天, 体积是地球体积的 8 倍, 行星表面的重力加速度是地球表面重力加速度的 2 倍, 它与中心恒星间的距离跟地球和太阳的距离相近。地球公转周期为 365 天，类地行星和地球均看作密度均匀的球体。求: 类地行星的中心恒星质量与太阳的质量之比 $\frac{M_{\text {恒 }}}{M_{\text {日 }}}$ 为多少?

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个数值。 问题: 某天体科学家在太阳系外发现一颗类地球行星, 这颗类地行星绕中心恒星做圆周运动, 公转的周期为 146 天, 体积是地球体积的 8 倍, 行星表面的重力加速度是地球表面重力加速度的 2 倍, 它与中心恒星间的距离跟地球和太阳的距离相近。地球公转周期为 365 天，类地行星和地球均看作密度均匀的球体。求: 类地行星的中心恒星质量与太阳的质量之比 $\frac{M_{\text {恒 }}}{M_{\text {日 }}}$ 为多少? 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是数值。

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Astronomy_460

质量相等的甲、乙两颗卫星分别贴近某星球表面和地球表面围绕其做匀速圆周运动,已知该星球和地球的密度相同, 半径分别为 $R$ 和 $r$, 则 ( ) A: 甲、乙两颗卫星的加速度之比等于 $R: r$ B: 甲、乙两颗卫星所受的向心力之比等于 $1: 1$ C: 甲、乙两颗卫星的线速度之比等于 $1: 1$ D: 甲、乙两颗卫星的周期之比等于 $1: 1$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 质量相等的甲、乙两颗卫星分别贴近某星球表面和地球表面围绕其做匀速圆周运动,已知该星球和地球的密度相同, 半径分别为 $R$ 和 $r$, 则 ( ) A: 甲、乙两颗卫星的加速度之比等于 $R: r$ B: 甲、乙两颗卫星所受的向心力之比等于 $1: 1$ C: 甲、乙两颗卫星的线速度之比等于 $1: 1$ D: 甲、乙两颗卫星的周期之比等于 $1: 1$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

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Astronomy_757

A comet's tail points in the following direction: A: away from the Sun B: towards the Sun C: in the direction of movement D: against the direction of movement

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: A comet's tail points in the following direction: A: away from the Sun B: towards the Sun C: in the direction of movement D: against the direction of movement You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

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Astronomy_999

What is the approximate average thickness of Saturn's rings? A: $10 \mu \mathrm{m}$ B: $10 \mathrm{~mm}$ C: $10 \mathrm{~m}$ D: $10 \mathrm{~km}$

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: What is the approximate average thickness of Saturn's rings? A: $10 \mu \mathrm{m}$ B: $10 \mathrm{~mm}$ C: $10 \mathrm{~m}$ D: $10 \mathrm{~km}$ You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

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