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GAIR 62

Astronomy_331

地球到太阳的距离为水星到太阳距离的 2.6 倍, 那么地球和水星绕太阳运行的线速度之比为 (设地球和水星绕太阳运行的轨道为圆)（） A: $\frac{1}{2.6}$ B: $\frac{2.6}{1}$ C: $\frac{1}{\sqrt{2.6}}$ D: $\frac{\sqrt{2.6}}{1}$

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: 地球到太阳的距离为水星到太阳距离的 2.6 倍, 那么地球和水星绕太阳运行的线速度之比为 (设地球和水星绕太阳运行的轨道为圆)（） A: $\frac{1}{2.6}$ B: $\frac{2.6}{1}$ C: $\frac{1}{\sqrt{2.6}}$ D: $\frac{\sqrt{2.6}}{1}$ 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

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Astronomy_771

Galileo discovered with his telescope that Venus ... A: has a diameter similar to the Earth B: has mountains on the surface. C: has phases like the Moon. D: has clouds in 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: Galileo discovered with his telescope that Venus ... A: has a diameter similar to the Earth B: has mountains on the surface. C: has phases like the Moon. D: has clouds in 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].

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Astronomy

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Astronomy_201

太阳系各行星几乎在同一平面内沿同一方向绕太阳做圆周运动。当地球恰好运行到某地外行星和太阳之间, 且三者几乎排成一条直线时, 天文学称这种现象为“行星冲日”。已知 2020 年 7 月 21 日土星冲日, 土星绕太阳运动的轨道半径约为地球绕太阳运动的轨道半径的 9.5 倍, 则下一次土星冲日的时间约为 ( ) A: 2021 年 8 月 B: 2022 年 7 月 C: 2023 年 8 月 D: 2024 年 7 月

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: 太阳系各行星几乎在同一平面内沿同一方向绕太阳做圆周运动。当地球恰好运行到某地外行星和太阳之间, 且三者几乎排成一条直线时, 天文学称这种现象为“行星冲日”。已知 2020 年 7 月 21 日土星冲日, 土星绕太阳运动的轨道半径约为地球绕太阳运动的轨道半径的 9.5 倍, 则下一次土星冲日的时间约为 ( ) A: 2021 年 8 月 B: 2022 年 7 月 C: 2023 年 8 月 D: 2024 年 7 月 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_518

2018 年 12 月 8 日中国在西昌卫星发射中心成功发射了嫦娥四号探测器, 经过地月转移飞行进入环月椭圆轨道, 然后实施近月制动, 顺利完成“太空刹车”, 被月球捕获,进入距月球表面约 $100 \mathrm{~km}$ 的环月圆形轨道, 准备登陆月球背面。如图所示, 关于嫦娥四号在环月椭圆轨道和环月圆形轨道的运行, 下列说法正确的是 ( ) [图1] A: 在环月粗圆轨道运行时, $A$ 点速度小于 $B$ 点速度 B: 由环月椭圆轨道进入环月圆形轨道, 嫦娥四号的机械能减小 C: 在环月制圆轨道上通过 $A$ 点的加速度比在环月圆形轨道通过 $A$ 点的加速度大 D: 在环月椭圆轨道运行的周期比环月圆形轨道的周期小

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: 2018 年 12 月 8 日中国在西昌卫星发射中心成功发射了嫦娥四号探测器, 经过地月转移飞行进入环月椭圆轨道, 然后实施近月制动, 顺利完成“太空刹车”, 被月球捕获,进入距月球表面约 $100 \mathrm{~km}$ 的环月圆形轨道, 准备登陆月球背面。如图所示, 关于嫦娥四号在环月椭圆轨道和环月圆形轨道的运行, 下列说法正确的是 ( ) [图1] A: 在环月粗圆轨道运行时, $A$ 点速度小于 $B$ 点速度 B: 由环月椭圆轨道进入环月圆形轨道, 嫦娥四号的机械能减小 C: 在环月制圆轨道上通过 $A$ 点的加速度比在环月圆形轨道通过 $A$ 点的加速度大 D: 在环月椭圆轨道运行的周期比环月圆形轨道的周期小 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_56

2021 年 5 月 15 日 7 时 18 分, 我国火星探测器“天问一号”的着陆巡视器 (其中巡视器就是“祝融号”火星车) 成功着陆于火星乌托邦平原南部预选着陆区, 这标志着我国是继美国、前苏联之后第三个成功进行火星探测的国家, 展示了中国在航天技术领域的强大实力, 为中国乃至国际航天事业迈出了历史性的一大步。“天问一号”的着陆巡视器从进入火星大气层到成功着陆经历了气动减速段、伞系减速段、动力减速段、悬停避障与缓速下降段, 其过程大致如图所示。已知火星质量为 $6.42 \times 10^{23} \mathrm{~kg}$ (约为地球质量的 0.11 倍)、半径为 $3395 \mathrm{~km}$ (约为地球半径的 0.53 倍), “天问一号”的着陆巡视器质量为 $1.3 \mathrm{t}$, 地球表面重力加速度为 $9.8 \mathrm{~m} / \mathrm{s}^{2}$ 。（计算）设着陆巡视器在伞系减速段做的是坚直方向的匀减速直线运动, 试求火星大气对着陆巡视器的平均阻力。(结果保留 3 位有效数字) [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数值。 问题: 2021 年 5 月 15 日 7 时 18 分, 我国火星探测器“天问一号”的着陆巡视器 (其中巡视器就是“祝融号”火星车) 成功着陆于火星乌托邦平原南部预选着陆区, 这标志着我国是继美国、前苏联之后第三个成功进行火星探测的国家, 展示了中国在航天技术领域的强大实力, 为中国乃至国际航天事业迈出了历史性的一大步。“天问一号”的着陆巡视器从进入火星大气层到成功着陆经历了气动减速段、伞系减速段、动力减速段、悬停避障与缓速下降段, 其过程大致如图所示。已知火星质量为 $6.42 \times 10^{23} \mathrm{~kg}$ (约为地球质量的 0.11 倍)、半径为 $3395 \mathrm{~km}$ (约为地球半径的 0.53 倍), “天问一号”的着陆巡视器质量为 $1.3 \mathrm{t}$, 地球表面重力加速度为 $9.8 \mathrm{~m} / \mathrm{s}^{2}$ 。（计算）设着陆巡视器在伞系减速段做的是坚直方向的匀减速直线运动, 试求火星大气对着陆巡视器的平均阻力。(结果保留 3 位有效数字) [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。 请记住，你的答案应以N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

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multi-modal

Astronomy_392

影片《流浪地球》中地球脱离太阳系流浪的最终目标是进入离太阳系最近的比邻星系的合适轨道, 成为这颗恒星的行星。现实中欧洲南方天文台曾宣布在离地球最近的比邻星发现宜居行星“比邻星 $b$ ”, 该行星质量约为地球的 1.3 倍, 直径约为地球的 22 倍,绕比邻星公转周期 11.2 天, 与比邻星距离约为日地距离的 5\%, 若不考虑星球的自转效应, 则 ( ) [图1] A: 比邻星的质量大于太阳的质量 B: 比邻星的密度小于太阳的密度 C: “比邻星 $b$ ”的公转线速度大小小于地球的公转线速度大小 D: “比邻星 $b$ ”表面的重力加速度小于地球表面的重力加速度

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 影片《流浪地球》中地球脱离太阳系流浪的最终目标是进入离太阳系最近的比邻星系的合适轨道, 成为这颗恒星的行星。现实中欧洲南方天文台曾宣布在离地球最近的比邻星发现宜居行星“比邻星 $b$ ”, 该行星质量约为地球的 1.3 倍, 直径约为地球的 22 倍,绕比邻星公转周期 11.2 天, 与比邻星距离约为日地距离的 5\%, 若不考虑星球的自转效应, 则 ( ) [图1] A: 比邻星的质量大于太阳的质量 B: 比邻星的密度小于太阳的密度 C: “比邻星 $b$ ”的公转线速度大小小于地球的公转线速度大小 D: “比邻星 $b$ ”表面的重力加速度小于地球表面的重力加速度 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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Astronomy

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multi-modal

Astronomy_859

With the technology currently available, it would take hundreds of millennia to send a humanmade object to other stars. A possible solution to this problem is to use relativistic light sails, which consist of very small probes propelled by radiation pressure. It is estimated that on the reference frame of an Earth observer, these sails would take 20.0 years to reach Alpha Centauri, which is 4.37 light-years away from the Solar System. The velocity of a light sail can be assumed to be constant throughout the entire trip. How long would this trip be on the reference frame of the light sail? A: 18.5 years B: 19.0 years C: 19.5 years D: 20.0 years E: 20.5 years

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: With the technology currently available, it would take hundreds of millennia to send a humanmade object to other stars. A possible solution to this problem is to use relativistic light sails, which consist of very small probes propelled by radiation pressure. It is estimated that on the reference frame of an Earth observer, these sails would take 20.0 years to reach Alpha Centauri, which is 4.37 light-years away from the Solar System. The velocity of a light sail can be assumed to be constant throughout the entire trip. How long would this trip be on the reference frame of the light sail? A: 18.5 years B: 19.0 years C: 19.5 years D: 20.0 years E: 20.5 years 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].

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Astronomy

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Astronomy_854

The spectrum of a blackbody peaks at a wavelength inversely proportional to its temperature. This is known as Wein's law and is used to estimate stellar temperatures. The sun can be approximated as a blackbody with its peak wavelength in the visible portion of the spectrum and a surface temperature of $6000 \mathrm{~K}$. Given this data, estimate the peak wavelength of a human being, assuming it to be a black body. A: $10 \mathrm{~nm}$ B: $10 \mu \mathrm{m}$ C: $10 \mathrm{~mm}$ D: $10 \mathrm{~m}$

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: The spectrum of a blackbody peaks at a wavelength inversely proportional to its temperature. This is known as Wein's law and is used to estimate stellar temperatures. The sun can be approximated as a blackbody with its peak wavelength in the visible portion of the spectrum and a surface temperature of $6000 \mathrm{~K}$. Given this data, estimate the peak wavelength of a human being, assuming it to be a black body. A: $10 \mathrm{~nm}$ B: $10 \mu \mathrm{m}$ C: $10 \mathrm{~mm}$ D: $10 \mathrm{~m}$ 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].

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Astronomy

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Astronomy_759

What is expected to happen at the end of the Sun's lifetime? A: Become a white dwarf B: Become a brown dwarf C: Explode as a supernova D: Become a neutron star or black hole

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 expected to happen at the end of the Sun's lifetime? A: Become a white dwarf B: Become a brown dwarf C: Explode as a supernova D: Become a neutron star or black hole 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_442

宇宙中两个相距较近的星球可以看成双星, 它们只在相互间的万有引力作用下, 绕二球心连线上的某一固定点做周期相同的匀速圆周运动. 根据宇宙大爆炸理论. 双星间的距离在不断缓慢增加, 设双星仍做匀速圆周运动, 则下列说法正确的是( A: 双星相互间的万有引力减小 B: 双星做圆周运动的角速度增大 C: 双星做圆周运动的周期减小 D: 双星做圆周运动的半径增大

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: 宇宙中两个相距较近的星球可以看成双星, 它们只在相互间的万有引力作用下, 绕二球心连线上的某一固定点做周期相同的匀速圆周运动. 根据宇宙大爆炸理论. 双星间的距离在不断缓慢增加, 设双星仍做匀速圆周运动, 则下列说法正确的是( A: 双星相互间的万有引力减小 B: 双星做圆周运动的角速度增大 C: 双星做圆周运动的周期减小 D: 双星做圆周运动的半径增大 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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Astronomy_846

The spectral line $H_{\alpha}$ in the spectrum of a star is recorded as having displacement of $\Delta \lambda=$ $0.043 \times 10^{-10} \mathrm{~m}$. At rest, the spectral line has a wavelength of $\lambda_{0}=6.563 \times 10^{-7} \mathrm{~m}$. Calculate the period of rotation for this star, if it is observed from its equatorial plane. We also know: $R_{\text {star }}=8 \times 10^{5} \mathrm{~km}$. A: 29.59 days B: 14.63 days C: 21.15 days D: 34.39 days

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 spectral line $H_{\alpha}$ in the spectrum of a star is recorded as having displacement of $\Delta \lambda=$ $0.043 \times 10^{-10} \mathrm{~m}$. At rest, the spectral line has a wavelength of $\lambda_{0}=6.563 \times 10^{-7} \mathrm{~m}$. Calculate the period of rotation for this star, if it is observed from its equatorial plane. We also know: $R_{\text {star }}=8 \times 10^{5} \mathrm{~km}$. A: 29.59 days B: 14.63 days C: 21.15 days D: 34.39 days 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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Astronomy_1110

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. iii. Since $q \propto \tau^{2} e^{-\tau}$ and $\tau \propto T^{-1 / 3}$, it can be approximated at a given temperature as $q \propto T^{\alpha}$, quantifying the sensitivity of the fusion reaction to temperature. By considering $d(\ln q) / d(\ln$ T) give an expression for $\alpha$ as a function of $\tau$ and calculate it at your central temperature.

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. iii. Since $q \propto \tau^{2} e^{-\tau}$ and $\tau \propto T^{-1 / 3}$, it can be approximated at a given temperature as $q \propto T^{\alpha}$, quantifying the sensitivity of the fusion reaction to temperature. By considering $d(\ln q) / d(\ln$ T) give an expression for $\alpha$ as a function of $\tau$ and calculate it at your central temperature. 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.

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Astronomy

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multi-modal

Astronomy_260

如图所示, $1 、 3$ 轨道均是卫星绕地球做圆周运动的轨道示意图, 1 轨道半径为 $R$, 2 轨道是一颗卫星绕地球做椭圆运动的轨道示意图, 3 轨道与 2 轨道相切于 $B$ 点, $O$ 点为地球球心, $A B$ 为椭圆的长轴, 两轨道和地心都在同一平面内。已知在 $1 、 2$ 两轨道上运动的卫星的周期相等, 万有引力常量为 $G$, 地球质量为 $M$, 三颗卫星的质量相等, 下列说法正确的是 ( ) [图1] A: 卫星在 3 轨道上的机械能小于在 2 轨道上的机械能 B: 若卫星在 I 轨道的速率为 $v_{l}$, 卫星在 2 轨道 $A$ 点的速率为 $\mathrm{v}_{\mathrm{A}}$, 则 $v_{l}<\mathrm{v}_{\mathrm{A}}$ C: 若卫星在 $1 、 3$ 轨道的加速度大小分别为 $a_{1} 、 a_{3}$, 卫星在 2 轨道 $B$ 点加速度大小为 $a_{B}$, 则 $a_{B}=a_{3}<a_{1}$ D: 若 $O A=0.4 R$, 则卫星在轨道 2 的 $B$ 点的速率 $v_{B}>\sqrt{\frac{5 G M}{8 R}}$

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: 如图所示, $1 、 3$ 轨道均是卫星绕地球做圆周运动的轨道示意图, 1 轨道半径为 $R$, 2 轨道是一颗卫星绕地球做椭圆运动的轨道示意图, 3 轨道与 2 轨道相切于 $B$ 点, $O$ 点为地球球心, $A B$ 为椭圆的长轴, 两轨道和地心都在同一平面内。已知在 $1 、 2$ 两轨道上运动的卫星的周期相等, 万有引力常量为 $G$, 地球质量为 $M$, 三颗卫星的质量相等, 下列说法正确的是 ( ) [图1] A: 卫星在 3 轨道上的机械能小于在 2 轨道上的机械能 B: 若卫星在 I 轨道的速率为 $v_{l}$, 卫星在 2 轨道 $A$ 点的速率为 $\mathrm{v}_{\mathrm{A}}$, 则 $v_{l}<\mathrm{v}_{\mathrm{A}}$ C: 若卫星在 $1 、 3$ 轨道的加速度大小分别为 $a_{1} 、 a_{3}$, 卫星在 2 轨道 $B$ 点加速度大小为 $a_{B}$, 则 $a_{B}=a_{3}<a_{1}$ D: 若 $O A=0.4 R$, 则卫星在轨道 2 的 $B$ 点的速率 $v_{B}>\sqrt{\frac{5 G M}{8 R}}$ 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_1025

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). 
Hence calculate the speed an astronaut would need to move at whilst travelling along its equator in order to keep the Sun in the same position in the sky. Assume Mercury has no axial tilt, and give your answer in $\mathrm{km} \mathrm{h}^{-1}$.

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). 
Hence calculate the speed an astronaut would need to move at whilst travelling along its equator in order to keep the Sun in the same position in the sky. Assume Mercury has no axial tilt, and give your answer in $\mathrm{km} \mathrm{h}^{-1}$. 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 km/h, 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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multi-modal

Astronomy_75

在星球 $M$ 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, $P$ 由静止向下运动, 物体的加速度 $a$ 与弹簧的压缩量 $x$ 间的关系如图中实线所示。在另一星球 $N$ 上用完全相同的弹簧, 改用物体 $Q$ 完成同样的过程, 其 $a-x$ 关系如图中虚线所示,假设两星球均为质量均匀分布的球体。已知星球 $M$ 的半径是星球 $N$ 的 3 倍，则（） [图1] A: $M$ 的质量是 $N$ 的 6 倍 B: $Q$ 的质量是 $P$ 的 6 倍 C: $M$ 与 $N$ 的密度相等 D: 贴近 $M$ 表面运行的卫星的周期是贴近 $N$ 表面运行的卫星的周期的 6 倍

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: 在星球 $M$ 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, $P$ 由静止向下运动, 物体的加速度 $a$ 与弹簧的压缩量 $x$ 间的关系如图中实线所示。在另一星球 $N$ 上用完全相同的弹簧, 改用物体 $Q$ 完成同样的过程, 其 $a-x$ 关系如图中虚线所示,假设两星球均为质量均匀分布的球体。已知星球 $M$ 的半径是星球 $N$ 的 3 倍，则（） [图1] A: $M$ 的质量是 $N$ 的 6 倍 B: $Q$ 的质量是 $P$ 的 6 倍 C: $M$ 与 $N$ 的密度相等 D: 贴近 $M$ 表面运行的卫星的周期是贴近 $N$ 表面运行的卫星的周期的 6 倍 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_234

2021 年 7 月 22 日上午 8 时我国首次公开了中国空间站核心舱组合体的运动轨道参数。现将其运行轨道简化为如图所示的绕地球 $O$ 运动的椭圆轨道, 地球位于制圆的一个焦点上, 其中 $A$ 为近地点, $B$ 为远地点。假设每隔 $\Delta t$ 时间记录一次核心舱的位置, 记录点如图所示, 已知 $E$ 为椭圆轨道的中心, $C 、 D 、 E$ 在同一条直线上且 $C D \perp A B, A B$的距离为 $2 a, C D$ 的距离为 $2 b$, 椭圆的面积公式为 $S=\pi a b$, 则核心舱从 $C$ 运动到 $B$ 所需的最短时间为 ( ) [图1] A: $\frac{7 \Delta t}{2}$ B: $\frac{7 \Delta t}{2}+\frac{7 \sqrt{a^{2}-b^{2}} \Delta t}{\pi a}$ C: $\frac{7 \Delta t}{2}+\frac{7 \sqrt{a^{2}-b^{2}} \Delta t}{\pi b}$ D: $12 \Delta t$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 2021 年 7 月 22 日上午 8 时我国首次公开了中国空间站核心舱组合体的运动轨道参数。现将其运行轨道简化为如图所示的绕地球 $O$ 运动的椭圆轨道, 地球位于制圆的一个焦点上, 其中 $A$ 为近地点, $B$ 为远地点。假设每隔 $\Delta t$ 时间记录一次核心舱的位置, 记录点如图所示, 已知 $E$ 为椭圆轨道的中心, $C 、 D 、 E$ 在同一条直线上且 $C D \perp A B, A B$的距离为 $2 a, C D$ 的距离为 $2 b$, 椭圆的面积公式为 $S=\pi a b$, 则核心舱从 $C$ 运动到 $B$ 所需的最短时间为 ( ) [图1] A: $\frac{7 \Delta t}{2}$ B: $\frac{7 \Delta t}{2}+\frac{7 \sqrt{a^{2}-b^{2}} \Delta t}{\pi a}$ C: $\frac{7 \Delta t}{2}+\frac{7 \sqrt{a^{2}-b^{2}} \Delta t}{\pi b}$ D: $12 \Delta t$ 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应为以下选项之一：[A, B, C, D]

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Astronomy

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multi-modal

Astronomy_624

2019 年 10 月 28 日发生了天王星冲日现象, 即太阳、地球、天王星处于同一直线,此时是观察天王星的最佳时间。已知日地距离为 $R_{0}$, 天王星和地球的公转周期分别为 $T$和 $T_{0}$, 则下列说法正确的是 ( ) A: 天王星与太阳的距离为 $\sqrt[3]{\frac{T^{2}}{T_{0}^{2}}} R_{0}$ B: 天王星与太阳的距离为 $\sqrt{\frac{T^{3}}{T_{0}^{3}}} R_{0}$ C: 至少再经过 $t=\frac{T T_{0}}{T-T_{0}}$ 时间, 天王星再次冲日 D: 至少再经过 $t=\frac{T T_{0}}{T+T_{0}}$ 时间，天王星再次冲日

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: 2019 年 10 月 28 日发生了天王星冲日现象, 即太阳、地球、天王星处于同一直线,此时是观察天王星的最佳时间。已知日地距离为 $R_{0}$, 天王星和地球的公转周期分别为 $T$和 $T_{0}$, 则下列说法正确的是 ( ) A: 天王星与太阳的距离为 $\sqrt[3]{\frac{T^{2}}{T_{0}^{2}}} R_{0}$ B: 天王星与太阳的距离为 $\sqrt{\frac{T^{3}}{T_{0}^{3}}} R_{0}$ C: 至少再经过 $t=\frac{T T_{0}}{T-T_{0}}$ 时间, 天王星再次冲日 D: 至少再经过 $t=\frac{T T_{0}}{T+T_{0}}$ 时间，天王星再次冲日 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_225

科学家通过研究双中子星合并的引力波, 发现: 两颗中子星在合并前相距为 $L$ 时,两者绕连线上的某点每秒转 $n$ 圈; 经过缓慢演化一段时间后, 两者的距离变为 $k L$, 每秒转 $p n$ 圈, 则演化前后 ( ) A: 两中子星运动周期为之前 $k p$ 倍 B: 两中子星运动的角速度为之前 $\frac{k}{p}$ 倍 C: 两中子星质量之和为之前 $k^{3} p^{2}$ 倍 D: 两中子星运动的线速度平方之和为之前 $\frac{1}{k}$ 倍

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: 科学家通过研究双中子星合并的引力波, 发现: 两颗中子星在合并前相距为 $L$ 时,两者绕连线上的某点每秒转 $n$ 圈; 经过缓慢演化一段时间后, 两者的距离变为 $k L$, 每秒转 $p n$ 圈, 则演化前后 ( ) A: 两中子星运动周期为之前 $k p$ 倍 B: 两中子星运动的角速度为之前 $\frac{k}{p}$ 倍 C: 两中子星质量之和为之前 $k^{3} p^{2}$ 倍 D: 两中子星运动的线速度平方之和为之前 $\frac{1}{k}$ 倍 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_900

Star formation begins when dense clumps within giant molecular clouds, called diffuse nebulae, begin to collapse as their gravity exceeds the pressure from the temperature of the gas. For much of the collapse the clump remains transparent and so can radiate away the energy of the collapse. Consequently the clump stays at a very constant temperature (until a protostar begins to form at the core, when the temperature does rise rapidly as the gas becomes opaque), and so this part of the collapse happens essentially in freefall. This means it can happen rather fast (by astronomical timescales!). [figure1] Figure 2: Left: Orion in visible light. Right: With radio detections of giant molecular clouds superimposed. Consider one of the very dense clumps within a nebula. Assume it is spherical with an initial radius of $r_{0}$, and begins to collapse at time $t=0$, until it has shrunk to radius $r$ by some time $t$ later. If the initial density of this clump, $\rho_{0}$, is uniform everywhere in the sphere (called a homologous collapse) then we can describe its collapse with the following formulae: $$ \theta+\frac{1}{2} \sin 2 \theta=\left(\frac{8 \pi G \rho_{0}}{3}\right)^{1 / 2} t \quad \text { where } \quad \frac{r}{r_{0}}=\cos ^{2} \theta $$ The duration of the freefall is $t_{\mathrm{ff}}$, and by the end of that part of the collapse $r \ll r_{0}$. Using the above equations derive an expression for $t_{\mathrm{ff}}$ in its simplest form. What do you notice about your formula? [Note: $\theta$ is in radians]

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is an expression. problem: Star formation begins when dense clumps within giant molecular clouds, called diffuse nebulae, begin to collapse as their gravity exceeds the pressure from the temperature of the gas. For much of the collapse the clump remains transparent and so can radiate away the energy of the collapse. Consequently the clump stays at a very constant temperature (until a protostar begins to form at the core, when the temperature does rise rapidly as the gas becomes opaque), and so this part of the collapse happens essentially in freefall. This means it can happen rather fast (by astronomical timescales!). [figure1] Figure 2: Left: Orion in visible light. Right: With radio detections of giant molecular clouds superimposed. Consider one of the very dense clumps within a nebula. Assume it is spherical with an initial radius of $r_{0}$, and begins to collapse at time $t=0$, until it has shrunk to radius $r$ by some time $t$ later. If the initial density of this clump, $\rho_{0}$, is uniform everywhere in the sphere (called a homologous collapse) then we can describe its collapse with the following formulae: $$ \theta+\frac{1}{2} \sin 2 \theta=\left(\frac{8 \pi G \rho_{0}}{3}\right)^{1 / 2} t \quad \text { where } \quad \frac{r}{r_{0}}=\cos ^{2} \theta $$ The duration of the freefall is $t_{\mathrm{ff}}$, and by the end of that part of the collapse $r \ll r_{0}$. Using the above equations derive an expression for $t_{\mathrm{ff}}$ in its simplest form. What do you notice about your formula? [Note: $\theta$ is in radians] 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_1220

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.e. If the disc has mass $M_{\text {disc }}=0.01 \mathrm{M} *$ and aspect ratio $h=0.05$, and if $q=5 \times 10^{-6}$ and $\tau_{0}=10$ years, find the elapsed time in years for the migration described in part $d$. to occur. Is this a feasible mechanism for planetary migration given the lifetime of the disc is roughly 10 Myr?

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: 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: e. If the disc has mass $M_{\text {disc }}=0.01 \mathrm{M} *$ and aspect ratio $h=0.05$, and if $q=5 \times 10^{-6}$ and $\tau_{0}=10$ years, find the elapsed time in years for the migration described in part $d$. to occur. Is this a feasible mechanism for planetary migration given the lifetime of the disc is roughly 10 Myr? 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{yrs}, 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_1039

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 $7.35 \times 10^{22} \mathrm{~kg}$.b. In reality, the effects of air resistance and the weight of the rocket are substantial. Once in the parking orbit it is travelling at $7.79 \mathrm{~km} \mathrm{~s}^{-1}$. ii. The Apollo 11 spacecraft was in the parking orbit for 2 hours 32 mins 27 secs. How many revolutions of the Earth did it do?

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 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: b. In reality, the effects of air resistance and the weight of the rocket are substantial. Once in the parking orbit it is travelling at $7.79 \mathrm{~km} \mathrm{~s}^{-1}$. ii. The Apollo 11 spacecraft was in the parking orbit for 2 hours 32 mins 27 secs. How many revolutions of the Earth did it do? 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_385

2007 年 4 月 24 日, 欧洲科学家宣布在太阳系之外发现了一颗可能适合人类居住的类地行星 Gliest581c.这颗围绕红矮星 Gliese 581 运行的星球有类似地球的温度，表面可能有液态水存在, 距离地球约为 20 光年, 直径约为地球的 1.5 倍, 质量约为地球的 5 倍, 绕红矮星 Gliese 581 运行的周期约为 13 天. 假设有一艘宇宙飞船飞临该星球表面附近轨道, 下列说法正确的是 ( ) A: 飞船在 Gliest $581 \mathrm{c}$ 表面附近运行的周期约为 13 天 B: 飞船在 Gliest $581 \mathrm{c}$ 表面附近运行时的速度大于 $7.9 \mathrm{~km} / \mathrm{s}$ C: 人在 Gliese $581 \mathrm{c}$ 上所受重力比在地球上所受重力大 D: Gliest $581 \mathrm{c}$ 的平均密度比地球平均密度小

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: 2007 年 4 月 24 日, 欧洲科学家宣布在太阳系之外发现了一颗可能适合人类居住的类地行星 Gliest581c.这颗围绕红矮星 Gliese 581 运行的星球有类似地球的温度，表面可能有液态水存在, 距离地球约为 20 光年, 直径约为地球的 1.5 倍, 质量约为地球的 5 倍, 绕红矮星 Gliese 581 运行的周期约为 13 天. 假设有一艘宇宙飞船飞临该星球表面附近轨道, 下列说法正确的是 ( ) A: 飞船在 Gliest $581 \mathrm{c}$ 表面附近运行的周期约为 13 天 B: 飞船在 Gliest $581 \mathrm{c}$ 表面附近运行时的速度大于 $7.9 \mathrm{~km} / \mathrm{s}$ C: 人在 Gliese $581 \mathrm{c}$ 上所受重力比在地球上所受重力大 D: Gliest $581 \mathrm{c}$ 的平均密度比地球平均密度小 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]

MC

Astronomy

ZH

text-only

Astronomy_504

2018 年 5 月 21 日, 我国发射人类首颗月球中继卫星“鹊桥”, 6 月 14 日进入使命轨道----地月拉格朗日 $L_{2}$ 轨道, 为在月球背面着陆的嫦娥四号与地球站之间提供通信链路。 12 月 8 日, 我国成功发射嫦娥四号探测器, 并于 2019 年 1 月 3 日成功着陆于与月球背面, 通过中继卫星“鹊桥"传回了月被影像图, 解开了古老月背的神秘面纱。如图所示, “鹊桥”中继星处于 $L_{2}$ 点上时, 会和月、地两个大天体保持相对静止的状态。设地球的质量为月球的 $k$ 倍，地月间距为 $L$, 拉格朗日 $L_{2}$ 点与月球间距为 $d$, 地球、月球和“鹊桥”均视为质点, 忽略太阳对“鹊桥”中继星的引力。则“鹊桥”中继星处于 $L_{2}$ 点上时, 下列选项正确的是 ( ) [图1] A: “鹊桥”与月球的线速度之比为 $v_{\text {穜 }}: v_{\text {月 }}=\sqrt{L}: \sqrt{L+d}$ B: “鹊桥”与月球的向心加速度之比为 $a_{\text {鹊 }}: a_{\text {月 }}=L:(L+d)$ C: $k, L, d$ 之间在关系为 $\frac{1}{(L+d)^{2}}+\frac{1}{k d^{2}}=\frac{L+d}{L^{3}}$ D: $k, L, d$ 之间在关系为 $\frac{1}{k(L+d)^{2}}+\frac{1}{d^{2}}=\frac{L+d}{L^{3}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 2018 年 5 月 21 日, 我国发射人类首颗月球中继卫星“鹊桥”, 6 月 14 日进入使命轨道----地月拉格朗日 $L_{2}$ 轨道, 为在月球背面着陆的嫦娥四号与地球站之间提供通信链路。 12 月 8 日, 我国成功发射嫦娥四号探测器, 并于 2019 年 1 月 3 日成功着陆于与月球背面, 通过中继卫星“鹊桥"传回了月被影像图, 解开了古老月背的神秘面纱。如图所示, “鹊桥”中继星处于 $L_{2}$ 点上时, 会和月、地两个大天体保持相对静止的状态。设地球的质量为月球的 $k$ 倍，地月间距为 $L$, 拉格朗日 $L_{2}$ 点与月球间距为 $d$, 地球、月球和“鹊桥”均视为质点, 忽略太阳对“鹊桥”中继星的引力。则“鹊桥”中继星处于 $L_{2}$ 点上时, 下列选项正确的是 ( ) [图1] A: “鹊桥”与月球的线速度之比为 $v_{\text {穜 }}: v_{\text {月 }}=\sqrt{L}: \sqrt{L+d}$ B: “鹊桥”与月球的向心加速度之比为 $a_{\text {鹊 }}: a_{\text {月 }}=L:(L+d)$ C: $k, L, d$ 之间在关系为 $\frac{1}{(L+d)^{2}}+\frac{1}{k d^{2}}=\frac{L+d}{L^{3}}$ D: $k, L, d$ 之间在关系为 $\frac{1}{k(L+d)^{2}}+\frac{1}{d^{2}}=\frac{L+d}{L^{3}}$ 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应为以下选项之一：[A, B, C, D]

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multi-modal

Astronomy_111

卫星 $\mathrm{M}$ 在轨道I上做匀速圆周运动, 一段时间后在 $A$ 点变速进入轨道II, 运行一段时间后, 在 $B$ 点变速进入轨道半径为轨道I轨道半径 5 倍的轨道III, 最后在轨道III做匀速圆周运动, 在轨道III上的速率为 $v$, 则卫星在轨道II上的 $B$ 点速率可能是 ( ) [图1] A: $\frac{1}{5} v$ B: $\frac{\sqrt{5}}{6} v$ C: $\frac{\sqrt{3}}{3} v$ D: $\frac{\sqrt{6}}{2} v$

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: 卫星 $\mathrm{M}$ 在轨道I上做匀速圆周运动, 一段时间后在 $A$ 点变速进入轨道II, 运行一段时间后, 在 $B$ 点变速进入轨道半径为轨道I轨道半径 5 倍的轨道III, 最后在轨道III做匀速圆周运动, 在轨道III上的速率为 $v$, 则卫星在轨道II上的 $B$ 点速率可能是 ( ) [图1] A: $\frac{1}{5} v$ B: $\frac{\sqrt{5}}{6} v$ C: $\frac{\sqrt{3}}{3} v$ D: $\frac{\sqrt{6}}{2} v$ 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_1122

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 $\gamma=2.194$. 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. The rate of change of frequency of the gravitational waves (called the 'chirp') from a merging binary can be written as $$ \dot{f}_{\mathrm{GW}}=\frac{96}{5} \pi^{8 / 3}\left(\frac{G \mathcal{M}}{c^{3}}\right)^{5 / 3} f_{\mathrm{GW}}^{11 / 3} $$d. Combine your result from $c$. with the above equation to cancel out $\mathcal{M}$ and so express the distance to the gravitational wave source, $r$, as a function of fundamental constants and the measurables $h, f_{G W}$, and $\dot{f}_{G W}$ only.

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: 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 $\gamma=2.194$. 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. The rate of change of frequency of the gravitational waves (called the 'chirp') from a merging binary can be written as $$ \dot{f}_{\mathrm{GW}}=\frac{96}{5} \pi^{8 / 3}\left(\frac{G \mathcal{M}}{c^{3}}\right)^{5 / 3} f_{\mathrm{GW}}^{11 / 3} $$ problem: d. Combine your result from $c$. with the above equation to cancel out $\mathcal{M}$ and so express the distance to the gravitational wave source, $r$, as a function of fundamental constants and the measurables $h, f_{G W}$, and $\dot{f}_{G W}$ only. 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

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Astronomy

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Astronomy_660

如图所示, 地球绕太阳做匀速圆周运动, 地球处在运动轨道 $b$ 位置时, 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角速度绕太阳做圆周运动, 下列说法正确的是 [图1] A: 若地球和该探测器分别在 $b 、 d$ 位置, 它们也能以相同的角速度绕太阳运动 B: 若地球和该探测器分别在 $b 、 e$ 位置, 它们也能以相同的角速度绕太阳运动 C: 该探测器在 $a$ 位置受到太阳、地球引力的合力等于在 $c$ 位置受到太阳, 地球引力的合力 D: 该探测器在 $a$ 位置受到太阳、地球引力的合力大于在 $c$ 位置受到太阳、地球引力的合力

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, 地球绕太阳做匀速圆周运动, 地球处在运动轨道 $b$ 位置时, 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角速度绕太阳做圆周运动, 下列说法正确的是 [图1] A: 若地球和该探测器分别在 $b 、 d$ 位置, 它们也能以相同的角速度绕太阳运动 B: 若地球和该探测器分别在 $b 、 e$ 位置, 它们也能以相同的角速度绕太阳运动 C: 该探测器在 $a$ 位置受到太阳、地球引力的合力等于在 $c$ 位置受到太阳, 地球引力的合力 D: 该探测器在 $a$ 位置受到太阳、地球引力的合力大于在 $c$ 位置受到太阳、地球引力的合力 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_248

万有引力作用下的物体具有引力势能, 取无穷远处引力势能为零, 物体距星球球心 距离为 $r$ 时的引力势能为 $E_{p}=-G \frac{M m}{r}$ ( $G$ 为引力常量, $M 、 m$ 分别为星球和物体的质量), 在一半径为 $R$ 的星球上, 一物体从星球表面某高度处自由下落(不计空气阻力),自开始下落计时, 得到物体在星球表面下落高度 $H$ 随时间 $t$ 变化的图象如图所示, 则 [图1] A: 在该星球表面上以 $\frac{1}{t_{0}} \sqrt{2 h R}$ 的初速度水平抛出一物体, 物体将不再落回星球表面 B: 在该星球表面上以 $\frac{2}{t_{0}} \sqrt{h R}$ 的初速度水平抛出一物体, 物体将不再落回星球表面 C: 在该星球表面上以 $\frac{1}{t_{0}} \sqrt{2 h R}$ 的初速度坚直上抛一物体, 物体将不再落回星球表面 D: 在该星球表面上以 $\frac{2}{t_{0}} \sqrt{h R}$ 的初速度坚直上抛一物体, 物体将不再落回星球表面

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: 万有引力作用下的物体具有引力势能, 取无穷远处引力势能为零, 物体距星球球心 距离为 $r$ 时的引力势能为 $E_{p}=-G \frac{M m}{r}$ ( $G$ 为引力常量, $M 、 m$ 分别为星球和物体的质量), 在一半径为 $R$ 的星球上, 一物体从星球表面某高度处自由下落(不计空气阻力),自开始下落计时, 得到物体在星球表面下落高度 $H$ 随时间 $t$ 变化的图象如图所示, 则 [图1] A: 在该星球表面上以 $\frac{1}{t_{0}} \sqrt{2 h R}$ 的初速度水平抛出一物体, 物体将不再落回星球表面 B: 在该星球表面上以 $\frac{2}{t_{0}} \sqrt{h R}$ 的初速度水平抛出一物体, 物体将不再落回星球表面 C: 在该星球表面上以 $\frac{1}{t_{0}} \sqrt{2 h R}$ 的初速度坚直上抛一物体, 物体将不再落回星球表面 D: 在该星球表面上以 $\frac{2}{t_{0}} \sqrt{h R}$ 的初速度坚直上抛一物体, 物体将不再落回星球表面 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

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multi-modal

Astronomy_701

已知地球自转周期为 $T_{0}$, 有一颗与同步卫星在同一轨道平面的低轨道卫星, 自西向东绕地球运行, 其运行半径为同步轨道半径的四分之一, 该卫星两次在同一城市的正上方出现的时间间隔可能是 A: $\frac{T_{0}}{4}$ B: $\frac{3 T_{0}}{7}$ C: $\frac{3 T_{0}}{4}$ D: $\frac{T_{0}}{7}$

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: 已知地球自转周期为 $T_{0}$, 有一颗与同步卫星在同一轨道平面的低轨道卫星, 自西向东绕地球运行, 其运行半径为同步轨道半径的四分之一, 该卫星两次在同一城市的正上方出现的时间间隔可能是 A: $\frac{T_{0}}{4}$ B: $\frac{3 T_{0}}{7}$ C: $\frac{3 T_{0}}{4}$ D: $\frac{T_{0}}{7}$ 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_106

4 月 29 日 11 时 23 分，在中国文昌航天发射场，长征五号 B 运载火箭将中国空间站工程首个航天器“天和”核心舱顺利送入太空, 任务取得圆满成功。未来两年, “天和” 将在距离地面约 $400 \mathrm{~km}$ 的圆轨道上，静候“天舟”“神舟”“问天”“梦天”等航天器的陆续来访, 共同完成空间站组装建造和关键技术在轨验证等建“宫”大业。假设地球的半径 $R=6400 \mathrm{~km}$ 。下列说法正确的是（ ） A: “天和”核心舱的发射速度小于 $7.9 \mathrm{~km} / \mathrm{s}$ B: “天和”核心舱的运行周期约等于 24 小时 C: “天和”核心舱的加速度约等于 $9 \mathrm{~m} / \mathrm{s}^{2}$ D: “天和”核心舱在圆轨道上运行速度大于 $7.9 \mathrm{~km} / \mathrm{s}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 4 月 29 日 11 时 23 分，在中国文昌航天发射场，长征五号 B 运载火箭将中国空间站工程首个航天器“天和”核心舱顺利送入太空, 任务取得圆满成功。未来两年, “天和” 将在距离地面约 $400 \mathrm{~km}$ 的圆轨道上，静候“天舟”“神舟”“问天”“梦天”等航天器的陆续来访, 共同完成空间站组装建造和关键技术在轨验证等建“宫”大业。假设地球的半径 $R=6400 \mathrm{~km}$ 。下列说法正确的是（ ） A: “天和”核心舱的发射速度小于 $7.9 \mathrm{~km} / \mathrm{s}$ B: “天和”核心舱的运行周期约等于 24 小时 C: “天和”核心舱的加速度约等于 $9 \mathrm{~m} / \mathrm{s}^{2}$ D: “天和”核心舱在圆轨道上运行速度大于 $7.9 \mathrm{~km} / \mathrm{s}$ 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应为以下选项之一：[A, B, C, D]

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Astronomy

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Astronomy_789

The Large Magellanic Cloud is .. A: a dwarf galaxy orbiting the Milky Way. B: the closest planetary nebula to the Earth. C: a bright star cluster discovered by Magellan. D: the outer arm of the Milky Way named after Magellan.

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 Large Magellanic Cloud is .. A: a dwarf galaxy orbiting the Milky Way. B: the closest planetary nebula to the Earth. C: a bright star cluster discovered by Magellan. D: the outer arm of the Milky Way named after Magellan. 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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Astronomy_282

我国发射的“天问一号”火星探测器到达火用后开展了一系列复杂的变轨操作: 2021 年 2 月 10 日, 探测器第一次到达近火点时被火星捕获, 成功实现火星环绕, 进入周期为 10 天的大椭圆轨道; 2 月 15 日, 探测器第一次到达远火点时进行变轨, 调整轨道平面与近火点高度, 环火轨道变为经过火星南北两极的极轨; 2 月 20 日, 探测器第二次到达近火点时进行轨道调整, 进入周期为 4 天的调相轨道; 2 月 24 日, 探测器第三次运行至近火点时顺利实施第三次近火制动, 成功进入停泊轨道。极轨、调相轨道、停泊轨道在同一平面内。探测器在这四次变轨过程中 [图1] A: 沿大粗圆轨道经过远火点与变轨后在极轨上经过远火点的加速度方向垂直 B: 沿极轨到达近火点变轨时制动减速才能进入调相轨道 C: 沿极轨、调相轨道经过近火点时的加速度都相等 D: 大椭圆轨道半长轴 $r_{1}$ 与调相轨道半长轴 $r_{2}$ 的比值为 $\frac{\sqrt[3]{400}}{4}$

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: 我国发射的“天问一号”火星探测器到达火用后开展了一系列复杂的变轨操作: 2021 年 2 月 10 日, 探测器第一次到达近火点时被火星捕获, 成功实现火星环绕, 进入周期为 10 天的大椭圆轨道; 2 月 15 日, 探测器第一次到达远火点时进行变轨, 调整轨道平面与近火点高度, 环火轨道变为经过火星南北两极的极轨; 2 月 20 日, 探测器第二次到达近火点时进行轨道调整, 进入周期为 4 天的调相轨道; 2 月 24 日, 探测器第三次运行至近火点时顺利实施第三次近火制动, 成功进入停泊轨道。极轨、调相轨道、停泊轨道在同一平面内。探测器在这四次变轨过程中 [图1] A: 沿大粗圆轨道经过远火点与变轨后在极轨上经过远火点的加速度方向垂直 B: 沿极轨到达近火点变轨时制动减速才能进入调相轨道 C: 沿极轨、调相轨道经过近火点时的加速度都相等 D: 大椭圆轨道半长轴 $r_{1}$ 与调相轨道半长轴 $r_{2}$ 的比值为 $\frac{\sqrt[3]{400}}{4}$ 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]

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multi-modal

Astronomy_46

开普勒发现了行星运动的三大定律, 分别是轨道定律、面积定律和周期定, 这三大定律最 终使他赢得了“天空立法者”的美名, 开普勒第一定律: 所有行星绕太阳运动的轨道都是椭圆, 太阳处在椭圆的一个焦点上. 开普勒第二定律：对任意一个行星来说,它与太阳的连线在相等的时间内扫过相等的面积 开普勒第三定律: 所有行星的轨道的半长轴的三次方跟它的公转周期的二次方的比值都相等即： $\frac{a^{3}}{T^{2}}=K$如图所示, 人造地球卫星在 I 轨道做匀速圆周运动时, 卫星距地面高度为 $h=3 R, R$为地球的半径, 卫星质量为 $m$, 地球表面的重力加速度为 $\mathrm{g}$, 椭圆轨道的长轴 $P Q=10 R$ 。 [图1]根据开普勒第三定律, 求卫星在II轨道运动时的周期大小;

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: 开普勒发现了行星运动的三大定律, 分别是轨道定律、面积定律和周期定, 这三大定律最 终使他赢得了“天空立法者”的美名, 开普勒第一定律: 所有行星绕太阳运动的轨道都是椭圆, 太阳处在椭圆的一个焦点上. 开普勒第二定律：对任意一个行星来说,它与太阳的连线在相等的时间内扫过相等的面积 开普勒第三定律: 所有行星的轨道的半长轴的三次方跟它的公转周期的二次方的比值都相等即： $\frac{a^{3}}{T^{2}}=K$如图所示, 人造地球卫星在 I 轨道做匀速圆周运动时, 卫星距地面高度为 $h=3 R, R$为地球的半径, 卫星质量为 $m$, 地球表面的重力加速度为 $\mathrm{g}$, 椭圆轨道的长轴 $P Q=10 R$ 。 [图1]根据开普勒第三定律, 求卫星在II轨道运动时的周期大小; 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

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multi-modal

Astronomy_79

2017 年, 人类第一次直接探测到来自双中子星合并的引力波。根据科学家们复原的过程, 在两颗中子星合并前约 $100 \mathrm{~s}$ 时, 它们相距约 $400 \mathrm{~km}$, 绕二者连线上的某点每秒转动 12 圈, 将两颗中子星都看作是质量均匀分布的球体, 由这些数据、万有引力常量并利用牛顿力学知识, 可以估算出这一时刻两颗中子星（） A: 质量之积 B: 质量之和 C: 速率之和 D: 各自的自转角速度

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 2017 年, 人类第一次直接探测到来自双中子星合并的引力波。根据科学家们复原的过程, 在两颗中子星合并前约 $100 \mathrm{~s}$ 时, 它们相距约 $400 \mathrm{~km}$, 绕二者连线上的某点每秒转动 12 圈, 将两颗中子星都看作是质量均匀分布的球体, 由这些数据、万有引力常量并利用牛顿力学知识, 可以估算出这一时刻两颗中子星（） A: 质量之积 B: 质量之和 C: 速率之和 D: 各自的自转角速度 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_637

预计于 2022 年建成的中国空间站将成为中国空间和新技术研究的重要基地。假设空间站中的宇航员利用电热器对食品加热, 电热器的加热线圈可以简化成如图甲所示的圆形闭合线圈, 其匝数为 $n$, 半径为 $r_{1}$, 总电阻为 $R_{0}$ 。将此线圈垂直放在圆形磁场中,且保证两圆心重合, 圆形磁场的半径为 $r_{2}\left(r_{2}>r_{1}\right)$, 磁感应强度大小 $B$ 随时间 $t$ 的变化关系如图乙所示。求: 在 $0 \sim 3 t_{0}$ 时间内线圈中产生的焦耳热; [图1] 图甲 [图2] 图乙

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: 预计于 2022 年建成的中国空间站将成为中国空间和新技术研究的重要基地。假设空间站中的宇航员利用电热器对食品加热, 电热器的加热线圈可以简化成如图甲所示的圆形闭合线圈, 其匝数为 $n$, 半径为 $r_{1}$, 总电阻为 $R_{0}$ 。将此线圈垂直放在圆形磁场中,且保证两圆心重合, 圆形磁场的半径为 $r_{2}\left(r_{2}>r_{1}\right)$, 磁感应强度大小 $B$ 随时间 $t$ 的变化关系如图乙所示。求: 在 $0 \sim 3 t_{0}$ 时间内线圈中产生的焦耳热; [图1] 图甲 [图2] 图乙 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_142

我国“天宫一号”圆满完成相关科学实验, 于 2018 年“受控”坠落. 若某航天器变轨后仍绕地球做匀速圆周运动, 但动能增大为原来的 4 倍, 不考虑航天器质量的变化, 则变轨后, 下列说法正确的是 ( ) A: 航天器的轨道半径变为原来的 $1 / 4$ B: 航天器的向心加速度变为原来的 4 倍 C: 航天器的周期变为原来的 $1 / 4$ D: 航天器的角速度变为原来的 4 倍

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“天宫一号”圆满完成相关科学实验, 于 2018 年“受控”坠落. 若某航天器变轨后仍绕地球做匀速圆周运动, 但动能增大为原来的 4 倍, 不考虑航天器质量的变化, 则变轨后, 下列说法正确的是 ( ) A: 航天器的轨道半径变为原来的 $1 / 4$ B: 航天器的向心加速度变为原来的 4 倍 C: 航天器的周期变为原来的 $1 / 4$ D: 航天器的角速度变为原来的 4 倍 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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text-only

Astronomy_793

Let $T_{\odot, C}$ and $T_{\odot, S}$ be the temperatures at the core and the surface of the sun, respectively. Similarly, let $T_{A, C}$ and $T_{A, S}$ be the temperatures at the core and surface of the red giant Arcturus, and let $T_{S, C}$ and $T_{S, S}$ be the temperatures at the core and surface of the white dwarf Sirius B. Which of the following inequalities is true? A: $\frac{T_{\odot, C}}{T_{\odot, S}}<\frac{T_{A, C}}{T_{A, S}}<\frac{T_{S, C}}{T_{S, S}}$ B: $\frac{T_{\odot, C}}{T_{\odot, S}}<\frac{T_{S, C}}{T_{S, S}}<\frac{T_{A, C}}{T_{A, S}}$ C: $\frac{T_{A, C}}{T_{A, S}}<\frac{T_{\odot, C}}{T_{\odot, S}}<\frac{T_{S, C}}{T_{S, S}}$ D: $\frac{T_{S, C}}{T_{S, S}}<\frac{T_{\odot, C}}{T_{\odot, S}}<\frac{T_{A, C}}{T_{A, S}}$ E: $\frac{T_{S, C}}{T_{S, S}}<\frac{T_{A, C}}{T_{A, S}}<\frac{T_{\odot, C}}{T_{\odot, S}}$

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: Let $T_{\odot, C}$ and $T_{\odot, S}$ be the temperatures at the core and the surface of the sun, respectively. Similarly, let $T_{A, C}$ and $T_{A, S}$ be the temperatures at the core and surface of the red giant Arcturus, and let $T_{S, C}$ and $T_{S, S}$ be the temperatures at the core and surface of the white dwarf Sirius B. Which of the following inequalities is true? A: $\frac{T_{\odot, C}}{T_{\odot, S}}<\frac{T_{A, C}}{T_{A, S}}<\frac{T_{S, C}}{T_{S, S}}$ B: $\frac{T_{\odot, C}}{T_{\odot, S}}<\frac{T_{S, C}}{T_{S, S}}<\frac{T_{A, C}}{T_{A, S}}$ C: $\frac{T_{A, C}}{T_{A, S}}<\frac{T_{\odot, C}}{T_{\odot, S}}<\frac{T_{S, C}}{T_{S, S}}$ D: $\frac{T_{S, C}}{T_{S, S}}<\frac{T_{\odot, C}}{T_{\odot, S}}<\frac{T_{A, C}}{T_{A, S}}$ E: $\frac{T_{S, C}}{T_{S, S}}<\frac{T_{A, C}}{T_{A, S}}<\frac{T_{\odot, C}}{T_{\odot, S}}$ 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].

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Astronomy

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Astronomy_1004

A comet orbits the Sun with a period of 172 years and eccentricity 0.94 . It is currently at a distance of 60 au away from the Sun. After which of these times will the comet be moving the fastest? A: 43 years B: 86 years C: 129 years D: 172 years

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 orbits the Sun with a period of 172 years and eccentricity 0.94 . It is currently at a distance of 60 au away from the Sun. After which of these times will the comet be moving the fastest? A: 43 years B: 86 years C: 129 years D: 172 years 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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Astronomy_728

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（图中 $a_{0}$未知）。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。 求星球 $\mathrm{A}$ 和星球 $\mathrm{B}$ 的表面重力加速度的比值; [图1] 图(a) [图2] 图(b)

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: 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（图中 $a_{0}$未知）。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。 求星球 $\mathrm{A}$ 和星球 $\mathrm{B}$ 的表面重力加速度的比值; [图1] 图(a) [图2] 图(b) 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

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multi-modal

Astronomy_21

已成为我国首个人造太阳系小行星的“嫦娥二号”，2014 年 2 月再次刷新我国深空探测最远距离纪录, 超过 7000 万公里, “嫦娥二号”是我国探月工程二期的先导星, 它先在距月球表面高度为 $h$ 的轨道上做匀速圆周运动, 运行周期为 $T$; 然后从月球轨道出发飞赴日地拉格朗日 $L_{2}$ 点（物体在该点受日、地引力平衡）进行科学探测。若以 $R$ 表示月球的半径, 引力常量为 $G$, 则 ( ) A: “嫦娥二号”卫星绕月运行时的线速度为 $\frac{2 \pi R}{T}$ B: 月球的质量为 $\frac{4 \pi^{2}(R+h)^{3}}{G T^{2}}$ C: 物体在月球表面自由下落的加速度为 $\frac{4 \pi^{2} R}{T^{2}}$ D: 嫦娥二号卫星在月球轨道需经过加速才能飞赴日地拉格朗日 $L_{2}$ 点

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: 已成为我国首个人造太阳系小行星的“嫦娥二号”，2014 年 2 月再次刷新我国深空探测最远距离纪录, 超过 7000 万公里, “嫦娥二号”是我国探月工程二期的先导星, 它先在距月球表面高度为 $h$ 的轨道上做匀速圆周运动, 运行周期为 $T$; 然后从月球轨道出发飞赴日地拉格朗日 $L_{2}$ 点（物体在该点受日、地引力平衡）进行科学探测。若以 $R$ 表示月球的半径, 引力常量为 $G$, 则 ( ) A: “嫦娥二号”卫星绕月运行时的线速度为 $\frac{2 \pi R}{T}$ B: 月球的质量为 $\frac{4 \pi^{2}(R+h)^{3}}{G T^{2}}$ C: 物体在月球表面自由下落的加速度为 $\frac{4 \pi^{2} R}{T^{2}}$ D: 嫦娥二号卫星在月球轨道需经过加速才能飞赴日地拉格朗日 $L_{2}$ 点 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_128

开普勒第三定律指出: 所有行星轨道的半长轴的三次方跟它的公转周期的二次方的比值都相等, 即 $\frac{a^{3}}{T^{2}}=k$, 其中 $a$ 表示椭圆轨道半长轴, $T$ 表示公转周期, 比值 $k$ 是一个对所有行星都相同的常量。同时, 开普勒第三定律对于轨迹为圆形和直线的运动依然适用: 圆形轨迹可以认为中心天体在圆心处, 半长轴为轨迹半径; 直线轨迹可以看成无限扁的椭圆轨迹, 长轴为物体与星球之间的距离。已知: 星球质量为 $M$, 在距离星球的距离为 $r$ 处有一物体, 该物体仅在星球引力的作用下运动。星球可视为质点且认为保持静止, 引力常量为 $G$, 则下列说法正确的是 ( ) A: 该星球和物体的引力系统中常量 $k=\frac{4 \pi^{2}}{G M}$ B: 要使物体绕星球做匀速圆周运动, 则物体的速度为 $v=\sqrt{\frac{2 G M}{r}}$ C: 若物体绕星球沿粗圆轨道运动, 在靠近星球的过程中动能在减少 D: 若物体由静止开始释放, 则该物体到达星球所经历的时间为 $t=\pi r \sqrt{\frac{r}{8 G M}}$

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: 开普勒第三定律指出: 所有行星轨道的半长轴的三次方跟它的公转周期的二次方的比值都相等, 即 $\frac{a^{3}}{T^{2}}=k$, 其中 $a$ 表示椭圆轨道半长轴, $T$ 表示公转周期, 比值 $k$ 是一个对所有行星都相同的常量。同时, 开普勒第三定律对于轨迹为圆形和直线的运动依然适用: 圆形轨迹可以认为中心天体在圆心处, 半长轴为轨迹半径; 直线轨迹可以看成无限扁的椭圆轨迹, 长轴为物体与星球之间的距离。已知: 星球质量为 $M$, 在距离星球的距离为 $r$ 处有一物体, 该物体仅在星球引力的作用下运动。星球可视为质点且认为保持静止, 引力常量为 $G$, 则下列说法正确的是 ( ) A: 该星球和物体的引力系统中常量 $k=\frac{4 \pi^{2}}{G M}$ B: 要使物体绕星球做匀速圆周运动, 则物体的速度为 $v=\sqrt{\frac{2 G M}{r}}$ C: 若物体绕星球沿粗圆轨道运动, 在靠近星球的过程中动能在减少 D: 若物体由静止开始释放, 则该物体到达星球所经历的时间为 $t=\pi r \sqrt{\frac{r}{8 G M}}$ 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_374

下面是地球、火星的有关情况比较。根据以上信息, 关于地球及火星（行星的运动 | 星球 | 地球 | 火星 | | :---: | :---: | :---: | | 公转半径 | $1.5 \times 10^{8} \mathrm{~km}$ | $2.25 \times$ <br> $10^{8} \mathrm{~km}$ | | 自转周期 | 23 时 56 分 | 24 时 37 分 | | 表面温度 | $15^{\circ} \mathrm{C}$ | $-100^{\circ} \mathrm{C} \sim 0^{\circ} \mathrm{C}$ | | 大气主要成分 | $78 \%$ 的 $\mathrm{N}_{2}, 21$ <br> $\%$ 的 $\mathrm{O}_{2}$ | 约 $95 \%$ 的 <br> $\mathrm{CO}_{2}$ | 可看做圆周运动 ), 下列推测正确的是（） A: 地球公转的线速度小于火星公转的线速度 B: 地球公转的向心加速度大于火星公转的向心加速度 C: 地球的自转角速度小于火星的自转角速度 D: 地球表面的重力加速度大于火星表面的重力加速度

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 下面是地球、火星的有关情况比较。根据以上信息, 关于地球及火星（行星的运动 | 星球 | 地球 | 火星 | | :---: | :---: | :---: | | 公转半径 | $1.5 \times 10^{8} \mathrm{~km}$ | $2.25 \times$ <br> $10^{8} \mathrm{~km}$ | | 自转周期 | 23 时 56 分 | 24 时 37 分 | | 表面温度 | $15^{\circ} \mathrm{C}$ | $-100^{\circ} \mathrm{C} \sim 0^{\circ} \mathrm{C}$ | | 大气主要成分 | $78 \%$ 的 $\mathrm{N}_{2}, 21$ <br> $\%$ 的 $\mathrm{O}_{2}$ | 约 $95 \%$ 的 <br> $\mathrm{CO}_{2}$ | 可看做圆周运动 ), 下列推测正确的是（） A: 地球公转的线速度小于火星公转的线速度 B: 地球公转的向心加速度大于火星公转的向心加速度 C: 地球的自转角速度小于火星的自转角速度 D: 地球表面的重力加速度大于火星表面的重力加速度 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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Astronomy_71

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: 神舟十五号在轨道 II 运动时从 $P$ 点运动到 $Q$ 点的最短时间; [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: 神舟十五号在轨道 II 运动时从 $P$ 点运动到 $Q$ 点的最短时间; [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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multi-modal

Astronomy_820

(CANCELED) When binary systems are really close together, they can execute an accretion process, in which one star (called the primary star) "eats" the mass of the other (called the secondary star), whose mass spirals down into the primary star, creating an accretion disk! For an accretion disk with the outer edge $3 R$ from the center of the primary star (radius $R$ and mass $M$ ), calculate the energy lost by a test mass (mass $m$ ) where it touches the primary star from where it first enters the accretion disk. Consider the orbits to be Keplerian. A: $\frac{G M m}{R}$ B: $\frac{1}{2} \frac{G M m}{R}$ C: $\frac{5}{2} \frac{G M m}{R}$ D: $\frac{1}{3} \frac{G M m}{R}$ E: $\frac{3}{4} \frac{G M m}{R}$

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: (CANCELED) When binary systems are really close together, they can execute an accretion process, in which one star (called the primary star) "eats" the mass of the other (called the secondary star), whose mass spirals down into the primary star, creating an accretion disk! For an accretion disk with the outer edge $3 R$ from the center of the primary star (radius $R$ and mass $M$ ), calculate the energy lost by a test mass (mass $m$ ) where it touches the primary star from where it first enters the accretion disk. Consider the orbits to be Keplerian. A: $\frac{G M m}{R}$ B: $\frac{1}{2} \frac{G M m}{R}$ C: $\frac{5}{2} \frac{G M m}{R}$ D: $\frac{1}{3} \frac{G M m}{R}$ E: $\frac{3}{4} \frac{G M m}{R}$ 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].

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Astronomy_497

'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: $Q$ 点处的重力加速度反常值; [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: $Q$ 点处的重力加速度反常值; [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_729

随着对宇宙的研究逐步开展, 科学家已多次探测到引力波。这证实了爱因斯坦 100 年前的预测，弥补了爱因斯坦广义相对论中最后一块缺失的“拼图”。双星的运动是产生引力波的来源之一，假设宇宙中有一由 $a 、 b$ 两颗星组成的双星系统，这两颗星在万有引力的作用下, 绕它们连线的某一点做匀速圆周运动, $a$ 星的运行周期为 $T, a 、 b$ 两颗星的距离为 $L, a 、 b$ 两颗星的轨道半径之差为 $\Delta r$ 。已知 $a$ 星的轨道半径大于 $b$ 星的轨道半径，则（） A: $b$ 星的周期为 $\frac{L-\Delta r}{L+\Delta r} T$ B: $b$ 星的线速度大小为 $\frac{\pi(L-\Delta r)}{T}$ C: $a 、 b$ 两颗星的半径之比为 $\frac{(L+\Delta r)^{2}}{(L-\Delta r)^{2}}$ D: $a 、 b$ 两颗星的质量之比为 $\frac{L-\Delta r}{L+\Delta r}$

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: 随着对宇宙的研究逐步开展, 科学家已多次探测到引力波。这证实了爱因斯坦 100 年前的预测，弥补了爱因斯坦广义相对论中最后一块缺失的“拼图”。双星的运动是产生引力波的来源之一，假设宇宙中有一由 $a 、 b$ 两颗星组成的双星系统，这两颗星在万有引力的作用下, 绕它们连线的某一点做匀速圆周运动, $a$ 星的运行周期为 $T, a 、 b$ 两颗星的距离为 $L, a 、 b$ 两颗星的轨道半径之差为 $\Delta r$ 。已知 $a$ 星的轨道半径大于 $b$ 星的轨道半径，则（） A: $b$ 星的周期为 $\frac{L-\Delta r}{L+\Delta r} T$ B: $b$ 星的线速度大小为 $\frac{\pi(L-\Delta r)}{T}$ C: $a 、 b$ 两颗星的半径之比为 $\frac{(L+\Delta r)^{2}}{(L-\Delta r)^{2}}$ D: $a 、 b$ 两颗星的质量之比为 $\frac{L-\Delta r}{L+\Delta r}$ 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]

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Astronomy_222

据报道, 已经发射成功的“嫦娥四号”月球探测器在月球背面实现了软着陆, 并展开探测工作, 它将通过早先发射的“鹊桥”中继卫星与地球实现信号传输及控制。在地月连线上存在一点“拉格朗日 $L_{2}$ ”, “鹊桥”在随月球绕地球同步公转的同时, 沿“Halo 轨道” (轨道平面与地月连线垂直) 绕 $L_{2}$ 转动, 如图所示。已知“鹊桥”卫星位于“Halo 轨道”时, 在地月引力共同作用下具有跟月球绕地球公转相同的周期。根据图中有关数据结合有关物理知识，可估算出 [图1] 地-月距离: $384400 \mathrm{~km}$ 鹊桥绕 $L_{2}$ 运转周期: $14 \mathrm{~d}$ 月- $L_{2}$ 距离: $64500 \mathrm{~km}$ Halo轨道半径: $3500 \mathrm{~km}$ 引力常量: $6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2}$ A: “鹊桥”质量 B: 月球质量 C: 月球的公转周期 D: “鹊桥”相对于 $L_{2}$ 的线速度

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 据报道, 已经发射成功的“嫦娥四号”月球探测器在月球背面实现了软着陆, 并展开探测工作, 它将通过早先发射的“鹊桥”中继卫星与地球实现信号传输及控制。在地月连线上存在一点“拉格朗日 $L_{2}$ ”, “鹊桥”在随月球绕地球同步公转的同时, 沿“Halo 轨道” (轨道平面与地月连线垂直) 绕 $L_{2}$ 转动, 如图所示。已知“鹊桥”卫星位于“Halo 轨道”时, 在地月引力共同作用下具有跟月球绕地球公转相同的周期。根据图中有关数据结合有关物理知识，可估算出 [图1] 地-月距离: $384400 \mathrm{~km}$ 鹊桥绕 $L_{2}$ 运转周期: $14 \mathrm{~d}$ 月- $L_{2}$ 距离: $64500 \mathrm{~km}$ Halo轨道半径: $3500 \mathrm{~km}$ 引力常量: $6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2}$ A: “鹊桥”质量 B: 月球质量 C: 月球的公转周期 D: “鹊桥”相对于 $L_{2}$ 的线速度 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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multi-modal

Astronomy_535

2022 年 4 月 16 日神舟十三号载人飞船返回舱成功着陆, 三位航天员在空间站出差半年, 完成了两次太空行走和 20 多项科学实验, 并开展了两次“天宫课堂”活动, 刷新了中国航天新纪录。已知地球半径为 $R$, 地球表面重力加速度为 $g$ 。总质量为 $m_{0}$ 的空间站绕地球的运动可近似为匀速圆周运动, 距地球表面高度为 $h$, 阻力忽略不计。 维持空间站的运行与舱内航天员的生活需要耗费大量电能, 某同学为其设计了太阳能电池板。太阳辐射的总功率为 $P_{0}$, 太阳与空间站的平均距离为 $r$, 且该太阳能电池板正对太阳的面积始终为 $S$, 假设该太阳能电池板的能量转化效率为 $\eta$ 。求单位时间空间站通过太阳能电池板获得的电能。

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表达式。 问题: 2022 年 4 月 16 日神舟十三号载人飞船返回舱成功着陆, 三位航天员在空间站出差半年, 完成了两次太空行走和 20 多项科学实验, 并开展了两次“天宫课堂”活动, 刷新了中国航天新纪录。已知地球半径为 $R$, 地球表面重力加速度为 $g$ 。总质量为 $m_{0}$ 的空间站绕地球的运动可近似为匀速圆周运动, 距地球表面高度为 $h$, 阻力忽略不计。 维持空间站的运行与舱内航天员的生活需要耗费大量电能, 某同学为其设计了太阳能电池板。太阳辐射的总功率为 $P_{0}$, 太阳与空间站的平均距离为 $r$, 且该太阳能电池板正对太阳的面积始终为 $S$, 假设该太阳能电池板的能量转化效率为 $\eta$ 。求单位时间空间站通过太阳能电池板获得的电能。 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。 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_1164

Some of the very first exoplanets to be discovered in large surveys were dubbed 'hot Jupiters' as they were similar in mass to Jupiter (i.e. a gas giant) but were much closer to their star than Mercury is to the Sun (and hence are in a very hot environment). Planetary formation models suggest that they were unlikely to have formed there, but instead formed much further out from the star and migrated inwards, due to gravitational interactions with other planets in the system. Studies of 'hot Jupiters' show that there is an overabundance of them with periods of $\sim 3-4$ days, and very few with periods shorter than that. Since large, close-in planets should be the easiest to detect in all of the main methods of finding exoplanets, this scarcity is likely to be a real effect and suggests that exoplanets which are that close to their star are in a relatively rapid (by astronomical standards) inspiral towards destruction by their star. [figure1] Figure 6: Left: The orbital radius of several 'hot Jupiters' scaled by the Roche radius of the system (where tidal forces would destroy the planet). There is an expected pile up close to radii double the Roche radius (dotted line), and very few with radii smaller than that - those that are will inevitably spiral into the star and be destroyed by the tidal forces when they get too close. Credit: Birkby et al. (2014). Right: As the planets inspiral we should see a shift in when their transits occur. This figure shows the predicted size of the shift after a period of 10 years if the tidal dissipation quality factor $Q_{\star}^{\prime}=10^{6}$, as well as the current detection limit of 5 seconds (dotted line). Therefore measuring if there is any shift in the transit times over the course of a decade of observations can put stringent limits on the value of $Q_{\star}^{\prime}$. Credit: Birkby et al. (2014). The Roche radius, where a planet will be torn apart due to the tidal forces acting on it, is defined as $$ a_{\text {Roche }} \approx 2.16 R_{P}\left(\frac{M_{\star}}{M_{P}}\right)^{1 / 3} $$ where $R_{P}$ is the radius of the planet, $M_{P}$ is the mass of the planet and $M_{\star}$ is the mass of the star. If a gas giant is knocked into a highly elliptical orbit (i.e. $e \approx 1$ ) that has a periapsis $r_{\text {peri }}<a_{\text {Roche }}$ then it will not survive. However, if the periapsis just grazes the Roche radius $\left(r_{\text {peri }} \approx a_{\text {Roche }}\right)$ then the orbit will rapidly circularise. By conserving angular momentum, it can be shown that the circular orbit will have a radius $a=2 a_{\text {Roche }}$ (see the left panel of Fig 6). Exoplanets observed to be in an orbit with a radius less than that will be unstable and angular momentum will be transferred from the planet to the star, causing the star to spin more rapidly and the planet's orbital radius to decrease. Eventually this will result in the planet's orbit crossing the Roche radius and being destroyed by the tidal forces. The duration of this inspiral will be dependent on how well the star can dissipate the orbital energy through frictional processes within the star, and can be parameterised by the tidal dissipation quality factor, $Q_{\star}^{\prime}$. By looking for changes in the orbital period of the planet, detectable by shifts in the timing of transits by the planet in front of the star, we can determine an estimate of $Q_{\star}^{\prime}$, which hence tells us about the internal structure of stars. These 'hot Jupiters' are the best laboratory we have for this, as they are the most likely to produce a measurable shift (i.e. $\sim 5 \mathrm{~s}$ ) in transit times within only $\sim 10$ years (see the right panel of Fig 6). We will try and reproduce these results in this question. The WTS-2 system consists of a star of mass $M_{\star}=0.820 M_{\odot}$, peak in its black-body spectrum at $\lambda_{\max }=580 \mathrm{~nm}$, and distance from us of $1.03 \mathrm{kpc}$, with an orbiting planet (called WTS-2b) with a period $P=1.0187$ days, mass of $1.12 M_{J}$ and radius $1.36 R_{J}$. The mass and radius of Jupiter are $M_{J}=1.90 \times 10^{27} \mathrm{~kg}$ and $R_{J}=7.15 \times 10^{7} \mathrm{~m}$ respectively. The change in the semi-major axis of the planet, $a$, due to tidal forces is given by $$ \left|\frac{\dot{a}}{a}\right|=6 k_{2} \Delta t \frac{M_{P}}{M_{\star}}\left(\frac{R_{\star}}{a}\right)^{5} n^{2} $$ where the dot notation is used to indicate the differential with respect to time (i.e. $\dot{a} \equiv \mathrm{d} a / \mathrm{d} t$ ), $k_{2}$ is a constant related to the density structure of the star, $\Delta t$ is the (assumed constant) time lag between where the planet is in its orbit and the location of the tidal bulge on the star, and $n=2 \pi / P$. By separating variables and integrating this equation, an expression can be derived for the time it takes for $a$ to decrease to zero. This is known as the inspiral time, $\tau$, and even though the planet will be destroyed when $a=a_{\text {Roche }}$ the time to go from $a=a_{\text {Roche }}$ to $a=0$ is negligible in comparison to the time to get to $a=a_{\text {Roche }}$, so $\tau$ is a good estimate of the remaining lifetime of the planet.b. Given the apparent magnitude of WTS-2 in the visible is $m=16.14$ and the absolute magnitude of the Sun in the same part of the EM spectrum is $\mathcal{M}_{\odot}=4.83$ : ii. Hence work out the radius of the star, $R_{\star}$. Give your answer in units of $R_{\odot}$.

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: Some of the very first exoplanets to be discovered in large surveys were dubbed 'hot Jupiters' as they were similar in mass to Jupiter (i.e. a gas giant) but were much closer to their star than Mercury is to the Sun (and hence are in a very hot environment). Planetary formation models suggest that they were unlikely to have formed there, but instead formed much further out from the star and migrated inwards, due to gravitational interactions with other planets in the system. Studies of 'hot Jupiters' show that there is an overabundance of them with periods of $\sim 3-4$ days, and very few with periods shorter than that. Since large, close-in planets should be the easiest to detect in all of the main methods of finding exoplanets, this scarcity is likely to be a real effect and suggests that exoplanets which are that close to their star are in a relatively rapid (by astronomical standards) inspiral towards destruction by their star. [figure1] Figure 6: Left: The orbital radius of several 'hot Jupiters' scaled by the Roche radius of the system (where tidal forces would destroy the planet). There is an expected pile up close to radii double the Roche radius (dotted line), and very few with radii smaller than that - those that are will inevitably spiral into the star and be destroyed by the tidal forces when they get too close. Credit: Birkby et al. (2014). Right: As the planets inspiral we should see a shift in when their transits occur. This figure shows the predicted size of the shift after a period of 10 years if the tidal dissipation quality factor $Q_{\star}^{\prime}=10^{6}$, as well as the current detection limit of 5 seconds (dotted line). Therefore measuring if there is any shift in the transit times over the course of a decade of observations can put stringent limits on the value of $Q_{\star}^{\prime}$. Credit: Birkby et al. (2014). The Roche radius, where a planet will be torn apart due to the tidal forces acting on it, is defined as $$ a_{\text {Roche }} \approx 2.16 R_{P}\left(\frac{M_{\star}}{M_{P}}\right)^{1 / 3} $$ where $R_{P}$ is the radius of the planet, $M_{P}$ is the mass of the planet and $M_{\star}$ is the mass of the star. If a gas giant is knocked into a highly elliptical orbit (i.e. $e \approx 1$ ) that has a periapsis $r_{\text {peri }}<a_{\text {Roche }}$ then it will not survive. However, if the periapsis just grazes the Roche radius $\left(r_{\text {peri }} \approx a_{\text {Roche }}\right)$ then the orbit will rapidly circularise. By conserving angular momentum, it can be shown that the circular orbit will have a radius $a=2 a_{\text {Roche }}$ (see the left panel of Fig 6). Exoplanets observed to be in an orbit with a radius less than that will be unstable and angular momentum will be transferred from the planet to the star, causing the star to spin more rapidly and the planet's orbital radius to decrease. Eventually this will result in the planet's orbit crossing the Roche radius and being destroyed by the tidal forces. The duration of this inspiral will be dependent on how well the star can dissipate the orbital energy through frictional processes within the star, and can be parameterised by the tidal dissipation quality factor, $Q_{\star}^{\prime}$. By looking for changes in the orbital period of the planet, detectable by shifts in the timing of transits by the planet in front of the star, we can determine an estimate of $Q_{\star}^{\prime}$, which hence tells us about the internal structure of stars. These 'hot Jupiters' are the best laboratory we have for this, as they are the most likely to produce a measurable shift (i.e. $\sim 5 \mathrm{~s}$ ) in transit times within only $\sim 10$ years (see the right panel of Fig 6). We will try and reproduce these results in this question. The WTS-2 system consists of a star of mass $M_{\star}=0.820 M_{\odot}$, peak in its black-body spectrum at $\lambda_{\max }=580 \mathrm{~nm}$, and distance from us of $1.03 \mathrm{kpc}$, with an orbiting planet (called WTS-2b) with a period $P=1.0187$ days, mass of $1.12 M_{J}$ and radius $1.36 R_{J}$. The mass and radius of Jupiter are $M_{J}=1.90 \times 10^{27} \mathrm{~kg}$ and $R_{J}=7.15 \times 10^{7} \mathrm{~m}$ respectively. The change in the semi-major axis of the planet, $a$, due to tidal forces is given by $$ \left|\frac{\dot{a}}{a}\right|=6 k_{2} \Delta t \frac{M_{P}}{M_{\star}}\left(\frac{R_{\star}}{a}\right)^{5} n^{2} $$ where the dot notation is used to indicate the differential with respect to time (i.e. $\dot{a} \equiv \mathrm{d} a / \mathrm{d} t$ ), $k_{2}$ is a constant related to the density structure of the star, $\Delta t$ is the (assumed constant) time lag between where the planet is in its orbit and the location of the tidal bulge on the star, and $n=2 \pi / P$. By separating variables and integrating this equation, an expression can be derived for the time it takes for $a$ to decrease to zero. This is known as the inspiral time, $\tau$, and even though the planet will be destroyed when $a=a_{\text {Roche }}$ the time to go from $a=a_{\text {Roche }}$ to $a=0$ is negligible in comparison to the time to get to $a=a_{\text {Roche }}$, so $\tau$ is a good estimate of the remaining lifetime of the planet. problem: b. Given the apparent magnitude of WTS-2 in the visible is $m=16.14$ and the absolute magnitude of the Sun in the same part of the EM spectrum is $\mathcal{M}_{\odot}=4.83$ : ii. Hence work out the radius of the star, $R_{\star}$. Give your answer in units of $R_{\odot}$. 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

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Astronomy

EN

multi-modal

Astronomy_918

One possible theory for why the gas giants have ring systems is that a small moon got too close to the parent planet. When the gravitational tidal forces (due to the difference between the strength of the planet's pull on the near and far sides of the moon) became greater than the gravitational forces holding the moon together, it was ripped apart. This minimum distance is called the "Roche limit", named after the French astronomer Edouard Roche who first calculated it. It is defined as when the gravitational force generated by the moon at its surface is equal to the tidal forces it experiences at that distance. [figure1] Consider a spherical planet with mass $M$ and radius $R$, and a perfectly rigid spherical moon with mass $m$ and radius $r$, orbiting the planet in a circular orbit of radius $d$. For a small particle of mass $u$ on the surface of the moon, the gravitational and tidal forces it experiences will be $$ F_{\text {grav }}=\frac{G m u}{r^{2}} \quad F_{\text {tidal }}=\frac{2 G M u r}{d^{3}} $$ Edouard Roche was one of the first scientists to suggest the destruction of a moon (which he named Veritas) as a source of Saturn's rings. Assume that it used to orbit in what is now the Cassini Division (an apparent gap in the rings at around $2 R_{\text {Saturn }}$ ), and that the fluid Roche limit was most relevant in this case. Given that the mass of the rings is $3.0 \times 10^{19} \mathrm{~kg}$, estimate the radius of Veritas.

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: One possible theory for why the gas giants have ring systems is that a small moon got too close to the parent planet. When the gravitational tidal forces (due to the difference between the strength of the planet's pull on the near and far sides of the moon) became greater than the gravitational forces holding the moon together, it was ripped apart. This minimum distance is called the "Roche limit", named after the French astronomer Edouard Roche who first calculated it. It is defined as when the gravitational force generated by the moon at its surface is equal to the tidal forces it experiences at that distance. [figure1] Consider a spherical planet with mass $M$ and radius $R$, and a perfectly rigid spherical moon with mass $m$ and radius $r$, orbiting the planet in a circular orbit of radius $d$. For a small particle of mass $u$ on the surface of the moon, the gravitational and tidal forces it experiences will be $$ F_{\text {grav }}=\frac{G m u}{r^{2}} \quad F_{\text {tidal }}=\frac{2 G M u r}{d^{3}} $$ Edouard Roche was one of the first scientists to suggest the destruction of a moon (which he named Veritas) as a source of Saturn's rings. Assume that it used to orbit in what is now the Cassini Division (an apparent gap in the rings at around $2 R_{\text {Saturn }}$ ), and that the fluid Roche limit was most relevant in this case. Given that the mass of the rings is $3.0 \times 10^{19} \mathrm{~kg}$, estimate the radius of Veritas. 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 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_829

An astronomer detected a galaxy and decided to analyze its different parts and physical aspects. The frequency generated by a "spin-flip" transition of atomic hydrogen is $\nu_{0}=1420.406 \mathrm{MHz}$, however it was detected on the galaxy as $\nu=1422.73$. He finds that: 9. Population I stars are (1) and are metal-(2). 10. The galaxy is (3) from us with a speed of (4) $k m * s^{-1}$. Choose the alternative that correctly completes sentences above. A: (1) young; (2) poor; (3) distancing; (4) 245 B: (1) old; (2) rich; (3) approaching; (4) 490 C: (1)old; (2) poor; (3) distancing; (4) 490 D: (1) young; (2) rich; (3) approaching; (4) 490 E: (1) young; (2) rich; (3) approaching; (4) 245

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 astronomer detected a galaxy and decided to analyze its different parts and physical aspects. The frequency generated by a "spin-flip" transition of atomic hydrogen is $\nu_{0}=1420.406 \mathrm{MHz}$, however it was detected on the galaxy as $\nu=1422.73$. He finds that: 9. Population I stars are (1) and are metal-(2). 10. The galaxy is (3) from us with a speed of (4) $k m * s^{-1}$. Choose the alternative that correctly completes sentences above. A: (1) young; (2) poor; (3) distancing; (4) 245 B: (1) old; (2) rich; (3) approaching; (4) 490 C: (1)old; (2) poor; (3) distancing; (4) 490 D: (1) young; (2) rich; (3) approaching; (4) 490 E: (1) young; (2) rich; (3) approaching; (4) 245 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_116

中国科技发展两个方向：“上天”和“入地”两大工程。其中，“上天”工程指“神舟”载人飞船、天宫空间站和探月工程; “入地”工程指“蛟龙”号深海下潜。若地球半径为 $R$,把地球看作质量分布均匀的球体 (已知质量分布均匀的球壳对球内任一质点的万有引力为零)。“天宫一号”轨道距离地面高度为 $h$, 所在处的重力加速度为 $g_{1}$; “蛟龙” 号下潜深度为 $d$, 所在处的重力加速度为 $g_{2}$; 地表处重力加速度为 $g$, 不计地球自转影响, 下列关系式正确的是（ ） A: $g_{1}=\frac{R}{R+h} g$ B: $g_{2}=\frac{R}{R-d} g$ C: $g_{1}=\frac{R^{3}}{(R-d)(R+h)^{2}} g_{2}$ D: $g_{1}=\frac{(R-d)(R+h)^{2}}{R^{3}} g_{2}$

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: 中国科技发展两个方向：“上天”和“入地”两大工程。其中，“上天”工程指“神舟”载人飞船、天宫空间站和探月工程; “入地”工程指“蛟龙”号深海下潜。若地球半径为 $R$,把地球看作质量分布均匀的球体 (已知质量分布均匀的球壳对球内任一质点的万有引力为零)。“天宫一号”轨道距离地面高度为 $h$, 所在处的重力加速度为 $g_{1}$; “蛟龙” 号下潜深度为 $d$, 所在处的重力加速度为 $g_{2}$; 地表处重力加速度为 $g$, 不计地球自转影响, 下列关系式正确的是（ ） A: $g_{1}=\frac{R}{R+h} g$ B: $g_{2}=\frac{R}{R-d} g$ C: $g_{1}=\frac{R^{3}}{(R-d)(R+h)^{2}} g_{2}$ D: $g_{1}=\frac{(R-d)(R+h)^{2}}{R^{3}} g_{2}$ 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_434

自“天空立法者”开普勒发现了开普勒三大定律后，人们对宇宙的探索从未停止。如图所示为行星 $P$ 的运行轨道, $F$ 为焦点（太阳）, $a 、 b 、 2 c$ 分别为半长轴, 半短轴和焦距, $O$ 为椭圆中心。根据万有引力定律, 行星和太阳间的引力势能为 $E_{\mathrm{p}}=-\frac{G M m}{r}$, 其中 $G$ 为引力常量, $M$ 为太阳的质量, $m$ 为行星的质量, $r$ 为太阳到行星的距离。行星 $P$在 $A 、 B$ 两点的瞬时速度为 $v_{A}$ 和 $v_{B}$, 下列说法正确的是（） [图1] A: $v_{B}=\frac{a-c}{a+c} v_{A}$ B: $P$ 在椭圆轨道的机械能为 $E=-G \frac{M m}{2 a}$ C: $P$ 在椭圆轨道的机械能为 $E=-G \frac{M m}{2 b}$ D: 行星 $P$ 绕 $F$ 运动的周期 $T=\frac{2 \pi a \sqrt{b}}{\sqrt{G M}}$

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: 自“天空立法者”开普勒发现了开普勒三大定律后，人们对宇宙的探索从未停止。如图所示为行星 $P$ 的运行轨道, $F$ 为焦点（太阳）, $a 、 b 、 2 c$ 分别为半长轴, 半短轴和焦距, $O$ 为椭圆中心。根据万有引力定律, 行星和太阳间的引力势能为 $E_{\mathrm{p}}=-\frac{G M m}{r}$, 其中 $G$ 为引力常量, $M$ 为太阳的质量, $m$ 为行星的质量, $r$ 为太阳到行星的距离。行星 $P$在 $A 、 B$ 两点的瞬时速度为 $v_{A}$ 和 $v_{B}$, 下列说法正确的是（） [图1] A: $v_{B}=\frac{a-c}{a+c} v_{A}$ B: $P$ 在椭圆轨道的机械能为 $E=-G \frac{M m}{2 a}$ C: $P$ 在椭圆轨道的机械能为 $E=-G \frac{M m}{2 b}$ D: 行星 $P$ 绕 $F$ 运动的周期 $T=\frac{2 \pi a \sqrt{b}}{\sqrt{G M}}$ 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_15

如图所示为 2022 年 11 月 8 日晚上出现的天文奇观“血月掩天王星”照片, 大大的月亮背景下天王星是一个小小的亮点。此时太阳、地球、月球、天王星几乎处于同一条直线上。已知地球和天王星绕太阳的公转方向与月球绕地球的公转方向相同, 下列说法正确的是 ( ) [图1] A: 在太阳参考系中, 此时地球的速度大于月球的速度 B: 在太阳参考系中, 此时地球的加速度小于月球的加速度 C: 下一次出现相同的天文奇观的时间间隔少于一年 D: 月球和天王星的公转轨道半长轴的三次方与公转周期的二次方的比值相等

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 2022 年 11 月 8 日晚上出现的天文奇观“血月掩天王星”照片, 大大的月亮背景下天王星是一个小小的亮点。此时太阳、地球、月球、天王星几乎处于同一条直线上。已知地球和天王星绕太阳的公转方向与月球绕地球的公转方向相同, 下列说法正确的是 ( ) [图1] A: 在太阳参考系中, 此时地球的速度大于月球的速度 B: 在太阳参考系中, 此时地球的加速度小于月球的加速度 C: 下一次出现相同的天文奇观的时间间隔少于一年 D: 月球和天王星的公转轨道半长轴的三次方与公转周期的二次方的比值相等 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_175

太阳系中的九大行星绕太阳公转的轨道均可视为圆, 不同行星的轨道平面均可视为同一平面. 如图所示, 当地球外侧的行星运动到日地连线上, 且和地球位于太阳同侧时,与地球的距离最近, 我们把这种相距最近的状态称为行星与地球的“会面”. 若每过 $N_{I}$年, 木星与地球“会面”一次, 每过 $N_{2}$ 年, 天王星与地球“会面”一次, 则木星与天王星 [图1] A: $\left[\frac{N_{1}\left(N_{2}-1\right)}{N_{2}\left(N_{1}-1\right)}\right]^{\frac{2}{3}}$ B: $\left[\frac{N_{2}\left(N_{1}-1\right)}{N_{1}\left(N_{2}-1\right)}\right]^{\frac{2}{3}}$ C: $\left[\frac{N_{1}\left(N_{1}-1\right)}{N_{2}\left(N_{2}-1\right)}\right]^{\frac{2}{3}}$ D: $\left[\frac{N_{2}\left(N_{2}-1\right)}{N_{1}\left(N_{1}-1\right)}\right]^{\frac{2}{3}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 太阳系中的九大行星绕太阳公转的轨道均可视为圆, 不同行星的轨道平面均可视为同一平面. 如图所示, 当地球外侧的行星运动到日地连线上, 且和地球位于太阳同侧时,与地球的距离最近, 我们把这种相距最近的状态称为行星与地球的“会面”. 若每过 $N_{I}$年, 木星与地球“会面”一次, 每过 $N_{2}$ 年, 天王星与地球“会面”一次, 则木星与天王星 [图1] A: $\left[\frac{N_{1}\left(N_{2}-1\right)}{N_{2}\left(N_{1}-1\right)}\right]^{\frac{2}{3}}$ B: $\left[\frac{N_{2}\left(N_{1}-1\right)}{N_{1}\left(N_{2}-1\right)}\right]^{\frac{2}{3}}$ C: $\left[\frac{N_{1}\left(N_{1}-1\right)}{N_{2}\left(N_{2}-1\right)}\right]^{\frac{2}{3}}$ D: $\left[\frac{N_{2}\left(N_{2}-1\right)}{N_{1}\left(N_{1}-1\right)}\right]^{\frac{2}{3}}$ 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应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

multi-modal

Astronomy_1090

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.iii. What would the receiving angle of each satellite's antenna need to be, and what would be the associated satellite footprint? By comparing these with the ones utilised by Earth's GPS, make a final comment on the viability of future Martian GPS.

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: iii. What would the receiving angle of each satellite's antenna need to be, and what would be the associated satellite footprint? By comparing these with the ones utilised by Earth's GPS, make a final comment on the viability of future Martian GPS. 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 \%, 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_313

2022 年 3 月 20 日 (太阳光直射赤道), 某颗地球同步卫星正下方的地球表面上有 一观察者, 他用天文望远镜观察被太阳光照射的此卫星, 在日落 12 小时内有 $t$ 时间该观察者看不见此卫星。若地球的质量为 $M$, 引力常量为 $G$, 地球半径为 $R$, 同步卫星离地高度为 $h$, 地球表面处的重力加速度为 $g$, 地球自转周期为 $T$, 卫星的绕地方向与地球自转方向相同, 不考虑大气对光的折射, 要计算出 $t$ 的数值, 则需要知道 ( ) A: $G 、 h 、 T$ 的数值 B: $R 、 h 、 T$ 的数值 C: $R 、 g 、 T$ 的数值 D: $R 、 M 、 T$ 的数值

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 3 月 20 日 (太阳光直射赤道), 某颗地球同步卫星正下方的地球表面上有 一观察者, 他用天文望远镜观察被太阳光照射的此卫星, 在日落 12 小时内有 $t$ 时间该观察者看不见此卫星。若地球的质量为 $M$, 引力常量为 $G$, 地球半径为 $R$, 同步卫星离地高度为 $h$, 地球表面处的重力加速度为 $g$, 地球自转周期为 $T$, 卫星的绕地方向与地球自转方向相同, 不考虑大气对光的折射, 要计算出 $t$ 的数值, 则需要知道 ( ) A: $G 、 h 、 T$ 的数值 B: $R 、 h 、 T$ 的数值 C: $R 、 g 、 T$ 的数值 D: $R 、 M 、 T$ 的数值 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_681

我国探月探测器“嫦娥三号”在西昌卫星发射中心成功发射升空, 此飞行轨道示意图如图所示, 探测器从地面发射后奔向月球, 在 $P$ 点从圆形轨道I进入粗圆轨道II, $Q$ 为轨道II上的近月点。下列关于“嫦娥三号”的运动, 正确的说法是 ( ) [图1] A: 在轨道II上经过 $P$ 的速度小于在轨道I上经过 $P$ 的速度 B: 在轨道II上经过 $P$ 的加速度小于在轨道I上经过 $P$ 的加速度 C: 发射速度一定大于 $7.9 \mathrm{~km} / \mathrm{s}$ D: 在轨道II上从 $P$ 到 $Q$ 的过程中速率不断增大

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: 我国探月探测器“嫦娥三号”在西昌卫星发射中心成功发射升空, 此飞行轨道示意图如图所示, 探测器从地面发射后奔向月球, 在 $P$ 点从圆形轨道I进入粗圆轨道II, $Q$ 为轨道II上的近月点。下列关于“嫦娥三号”的运动, 正确的说法是 ( ) [图1] A: 在轨道II上经过 $P$ 的速度小于在轨道I上经过 $P$ 的速度 B: 在轨道II上经过 $P$ 的加速度小于在轨道I上经过 $P$ 的加速度 C: 发射速度一定大于 $7.9 \mathrm{~km} / \mathrm{s}$ D: 在轨道II上从 $P$ 到 $Q$ 的过程中速率不断增大 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]

MC

Astronomy

ZH

multi-modal

Astronomy_886

What would be the radius of the Sun when it becomes a black hole (although the probability is almost zero)? A: $1 \mathrm{~km}$ B: $2 \mathrm{~km}$ C: $3 \mathbf{k m} D: $4 \mathrm{~km}$ E: $5 \mathrm{~km}$

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 would be the radius of the Sun when it becomes a black hole (although the probability is almost zero)? A: $1 \mathrm{~km}$ B: $2 \mathrm{~km}$ C: $3 \mathbf{k m} D: $4 \mathrm{~km}$ E: $5 \mathrm{~km}$ 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, E].

SC

Astronomy

EN

text-only

Astronomy_914

In which constellation is Saturn visible during Autumn 2021? [Hint: it is close to Jupiter, which spent most of Summer 2021 in Aquarius] A: Capricorn B: Gemini C: Leo D: Libra

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: In which constellation is Saturn visible during Autumn 2021? [Hint: it is close to Jupiter, which spent most of Summer 2021 in Aquarius] A: Capricorn B: Gemini C: Leo D: Libra 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_205

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, 在此轨道正常运行时, 卫星的轨道半径为 $\mathrm{R}_{1}$ 、周期为 $\mathrm{T}_{1}$ 、经过 $\mathrm{p}$ 点的速度大小为 $\mathrm{V}_{1}$ 、加速度大小为 $a_{1}$; 然后在 $\mathrm{P}$ 点点火加速, 进入粗圆形转移轨道II, 在此轨道正常运行时, 卫星的周期为 $\mathrm{T}_{2}$, 经过 $\mathrm{p}$ 点的速度大小为 $\mathrm{V}_{2}$ 、加速度大小为 $a_{2}$, 经过 $\mathrm{Q}$ 点速度大小为 $\mathrm{V}_{3}$; 稳定运行数圈后达远地点 $\mathrm{Q}$ 时再次点火加速, 进入远地圆轨道III在此轨道正常运行时, 卫星的轨道半径为 $\mathrm{R}_{3}$ 、周期为 $\mathrm{T}_{3}$ 、经过 $\mathrm{Q}$ 点速度大小为 $\mathrm{V}_{4}$ (轨道II的近地点和远地点分别为轨道 $I$ 上的 $P$ 点、轨道III上的 $Q$ 点). 已知 $R_{3}=2 R_{1}$, 则下列关系正确的是 $(\quad)$ [图1] A: $\mathrm{T}_{2}=3 \sqrt{3} \mathrm{~T}_{1}$ B: $\mathrm{T}_{2}=\frac{3 \sqrt{3}}{8} \mathrm{~T}_{3}$ C: $a_{1}=a_{2}$ D: $V_{2}>V_{1}>V_{4}>V_{3}$

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: 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, 在此轨道正常运行时, 卫星的轨道半径为 $\mathrm{R}_{1}$ 、周期为 $\mathrm{T}_{1}$ 、经过 $\mathrm{p}$ 点的速度大小为 $\mathrm{V}_{1}$ 、加速度大小为 $a_{1}$; 然后在 $\mathrm{P}$ 点点火加速, 进入粗圆形转移轨道II, 在此轨道正常运行时, 卫星的周期为 $\mathrm{T}_{2}$, 经过 $\mathrm{p}$ 点的速度大小为 $\mathrm{V}_{2}$ 、加速度大小为 $a_{2}$, 经过 $\mathrm{Q}$ 点速度大小为 $\mathrm{V}_{3}$; 稳定运行数圈后达远地点 $\mathrm{Q}$ 时再次点火加速, 进入远地圆轨道III在此轨道正常运行时, 卫星的轨道半径为 $\mathrm{R}_{3}$ 、周期为 $\mathrm{T}_{3}$ 、经过 $\mathrm{Q}$ 点速度大小为 $\mathrm{V}_{4}$ (轨道II的近地点和远地点分别为轨道 $I$ 上的 $P$ 点、轨道III上的 $Q$ 点). 已知 $R_{3}=2 R_{1}$, 则下列关系正确的是 $(\quad)$ [图1] A: $\mathrm{T}_{2}=3 \sqrt{3} \mathrm{~T}_{1}$ B: $\mathrm{T}_{2}=\frac{3 \sqrt{3}}{8} \mathrm{~T}_{3}$ C: $a_{1}=a_{2}$ D: $V_{2}>V_{1}>V_{4}>V_{3}$ 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]

MC

Astronomy

ZH

multi-modal

Astronomy_667

“信使号”水星探测器按计划将在今年陨落在水星表面。工程师通过向后释放推进系统中的高压氦气来提升轨道, 使其寿命再延长一个月, 如图所示, 释放氦气前, 探测器在贴近水星表面的圆形轨道I上做匀速圆周运动, 释放氦气后探测器进入粗圆轨道II, 忽略探测器在粗圆轨道上所受阻力, 则下列说法正确的是（） [图1] A: 探测器在轨道I上 $E$ 点速率大于在轨道II上 $E$ 点速率 B: 探测器在轨道II上任意位置的速率都大于在轨道I上速率 C: 探测器在轨道I和轨道II上的 $E$ 点处加速度不相同 D: 探测器在轨道II上远离水星过程中, 动能减少但势能增加

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: “信使号”水星探测器按计划将在今年陨落在水星表面。工程师通过向后释放推进系统中的高压氦气来提升轨道, 使其寿命再延长一个月, 如图所示, 释放氦气前, 探测器在贴近水星表面的圆形轨道I上做匀速圆周运动, 释放氦气后探测器进入粗圆轨道II, 忽略探测器在粗圆轨道上所受阻力, 则下列说法正确的是（） [图1] A: 探测器在轨道I上 $E$ 点速率大于在轨道II上 $E$ 点速率 B: 探测器在轨道II上任意位置的速率都大于在轨道I上速率 C: 探测器在轨道I和轨道II上的 $E$ 点处加速度不相同 D: 探测器在轨道II上远离水星过程中, 动能减少但势能增加 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_1060

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 $\gamma=2.194$. 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. The rate of change of frequency of the gravitational waves (called the 'chirp') from a merging binary can be written as $$ \dot{f}_{\mathrm{GW}}=\frac{96}{5} \pi^{8 / 3}\left(\frac{G \mathcal{M}}{c^{3}}\right)^{5 / 3} f_{\mathrm{GW}}^{11 / 3} $$a. The galaxy NGC 4993 is measured to have a redshift of $z=0.00980 \pm 0.00079$. Assuming it follows Hubble's Law, $v=H_{0} d$, where $H_{0}=73.24 \pm 1.74 \mathrm{~km} \mathrm{~s}^{-1} \mathrm{Mpc}^{-1}$, as determined by Hubble Space Telescope (HST) measurements of Cepheid variables, calculate the distance to the galaxy (in Mpc) and its (absolute) uncertainty. Give your distance to an appropriate number of significant figures.

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 $\gamma=2.194$. 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. The rate of change of frequency of the gravitational waves (called the 'chirp') from a merging binary can be written as $$ \dot{f}_{\mathrm{GW}}=\frac{96}{5} \pi^{8 / 3}\left(\frac{G \mathcal{M}}{c^{3}}\right)^{5 / 3} f_{\mathrm{GW}}^{11 / 3} $$ problem: a. The galaxy NGC 4993 is measured to have a redshift of $z=0.00980 \pm 0.00079$. Assuming it follows Hubble's Law, $v=H_{0} d$, where $H_{0}=73.24 \pm 1.74 \mathrm{~km} \mathrm{~s}^{-1} \mathrm{Mpc}^{-1}$, as determined by Hubble Space Telescope (HST) measurements of Cepheid variables, calculate the distance to the galaxy (in Mpc) and its (absolute) uncertainty. Give your distance to an appropriate number of significant figures. 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{Mpc}, 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

text-only

Astronomy_706

太空中存在一些离其它恒星较远的、由质量相等的三颗星组成的三星系统, 通常可忽略其它星体对它们的引力作用. 已观测到稳定的三星系统存在两种基本的构成形式: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 $\mathrm{R}$ 的圆轨道上运行; 另一种形式是三颗星位于等边三角形的三个顶点上, 并沿外接于等边三角形的圆形轨道运行. 设这三个星体的质量均为 $\mathrm{M}$, 并设两种系统的运动周期相同，则 （ ) [图1] A: 直线三星系统运动的线速度大小为 $v=\sqrt{\frac{G M}{R}}$ B: 三星系统的运动周期为 $T=4 \pi R \sqrt{\frac{R}{5 G M}}$ C: 三角形三星系统中星体间的距离为 $L=\sqrt[3]{\frac{12}{5}} R$ D: 三角形三星系统的线速度大小为 $v=\frac{1}{2} \sqrt{\frac{5 G M}{R}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 太空中存在一些离其它恒星较远的、由质量相等的三颗星组成的三星系统, 通常可忽略其它星体对它们的引力作用. 已观测到稳定的三星系统存在两种基本的构成形式: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 $\mathrm{R}$ 的圆轨道上运行; 另一种形式是三颗星位于等边三角形的三个顶点上, 并沿外接于等边三角形的圆形轨道运行. 设这三个星体的质量均为 $\mathrm{M}$, 并设两种系统的运动周期相同，则 （ ) [图1] A: 直线三星系统运动的线速度大小为 $v=\sqrt{\frac{G M}{R}}$ B: 三星系统的运动周期为 $T=4 \pi R \sqrt{\frac{R}{5 G M}}$ C: 三角形三星系统中星体间的距离为 $L=\sqrt[3]{\frac{12}{5}} R$ D: 三角形三星系统的线速度大小为 $v=\frac{1}{2} \sqrt{\frac{5 G M}{R}}$ 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应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_1148

The speed of light is considered to be the speed limit of the Universe, however knots of plasma in the jets from active galactic nuclei (AGN) have been observed to be moving with apparent transverse speeds in excess of this, called superluminal speeds. Some of the more extreme examples can be appearing to move at up to 6 times the speed of light (see Figure 7). [figure1] Figure 7: Left: The jet coming from the elliptical galaxy M87 as viewed by the Hubble Space Telescope (HST). Right: Sequence of HST images showing motion at six times the speed of light. The slanting lines track the moving features, and the speeds are given in units of the velocity of light, $c$. Credit: NASA / Space Telescope Science Institute / John Biretta. This can be explained by understanding that the jet is offset by an angle $\theta$ from the sightline to Earth, and that the real speed of the plasma knot, $v$, is less than $c$, and from it we can define the scaled speed $\beta \equiv v / c$. Superluminal jets are not limited just to AGN, as they have also been observed from systems within our own galaxy. A particularly famous one is the 'microquasar' GRS $1915+105$, which is a low mass X-ray binary consisting of a small star orbiting a black hole. A symmetrical jet with components approaching and receding from us is observed (as expected for jets coming from the poles of the black hole), and the apparent transverse motion of material in those jets has been measured using very high resolution radio imaging. Fender et. al (1999) measure these motions to be $\mu_{a}=23.6$ mas day $^{-1}$ and $\mu_{r}=$ 10.0 mas day $^{-1}$ for the approaching and receding jet respectively ( 1 mas $=1$ milliarcsecond, a unit of angle, and there are 3600 arcseconds in a degree) and the distance to the system as $11 \mathrm{kpc}$. In practice, for a given $\beta_{\text {app }}$ the values of $\beta$ and $\theta$ are degenerate and it is unlikely that the orientation of the jet is such that $\beta_{\text {app }}$ has been maximised, so the value in part $\mathrm{c}$. is just a lower limit. However, since there are two jets then if we assume that they are from the same event (and so equal in speed but opposite in direction) we can break this degeneracy. Since it is a binary system, we can gain information about the masses of the objects by looking at their period and radial velocity. Formally, the relationship is $$ \frac{\left(M_{\mathrm{BH}} \sin i\right)^{3}}{\left(M_{\mathrm{BH}}+M_{\star}\right)^{2}}=\frac{P_{\mathrm{orb}} K_{d}^{3}}{2 \pi G} $$ where $M_{\mathrm{BH}}$ is the mass of the black hole, $M_{\star}$ is the mass of the orbiting star, $i$ is the inclination of the orbit, $P_{\text {orb }}$ is the orbital period, and $K_{d}$ is the amplitude of the radial velocity curve. Normally the inclination can't be measured, however if we assume that the orbit is perpendicular to the jets then $i=\theta$ and we can measure the mass of the black hole.e. Show that $\beta \cos \theta$ can be expressed purely as a function of $\mu_{a}$ and $\mu_{r}$, and hence use your value of $\theta$ to calculate the value of $\beta$.

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 speed of light is considered to be the speed limit of the Universe, however knots of plasma in the jets from active galactic nuclei (AGN) have been observed to be moving with apparent transverse speeds in excess of this, called superluminal speeds. Some of the more extreme examples can be appearing to move at up to 6 times the speed of light (see Figure 7). [figure1] Figure 7: Left: The jet coming from the elliptical galaxy M87 as viewed by the Hubble Space Telescope (HST). Right: Sequence of HST images showing motion at six times the speed of light. The slanting lines track the moving features, and the speeds are given in units of the velocity of light, $c$. Credit: NASA / Space Telescope Science Institute / John Biretta. This can be explained by understanding that the jet is offset by an angle $\theta$ from the sightline to Earth, and that the real speed of the plasma knot, $v$, is less than $c$, and from it we can define the scaled speed $\beta \equiv v / c$. Superluminal jets are not limited just to AGN, as they have also been observed from systems within our own galaxy. A particularly famous one is the 'microquasar' GRS $1915+105$, which is a low mass X-ray binary consisting of a small star orbiting a black hole. A symmetrical jet with components approaching and receding from us is observed (as expected for jets coming from the poles of the black hole), and the apparent transverse motion of material in those jets has been measured using very high resolution radio imaging. Fender et. al (1999) measure these motions to be $\mu_{a}=23.6$ mas day $^{-1}$ and $\mu_{r}=$ 10.0 mas day $^{-1}$ for the approaching and receding jet respectively ( 1 mas $=1$ milliarcsecond, a unit of angle, and there are 3600 arcseconds in a degree) and the distance to the system as $11 \mathrm{kpc}$. In practice, for a given $\beta_{\text {app }}$ the values of $\beta$ and $\theta$ are degenerate and it is unlikely that the orientation of the jet is such that $\beta_{\text {app }}$ has been maximised, so the value in part $\mathrm{c}$. is just a lower limit. However, since there are two jets then if we assume that they are from the same event (and so equal in speed but opposite in direction) we can break this degeneracy. Since it is a binary system, we can gain information about the masses of the objects by looking at their period and radial velocity. Formally, the relationship is $$ \frac{\left(M_{\mathrm{BH}} \sin i\right)^{3}}{\left(M_{\mathrm{BH}}+M_{\star}\right)^{2}}=\frac{P_{\mathrm{orb}} K_{d}^{3}}{2 \pi G} $$ where $M_{\mathrm{BH}}$ is the mass of the black hole, $M_{\star}$ is the mass of the orbiting star, $i$ is the inclination of the orbit, $P_{\text {orb }}$ is the orbital period, and $K_{d}$ is the amplitude of the radial velocity curve. Normally the inclination can't be measured, however if we assume that the orbit is perpendicular to the jets then $i=\theta$ and we can measure the mass of the black hole. problem: e. Show that $\beta \cos \theta$ can be expressed purely as a function of $\mu_{a}$ and $\mu_{r}$, and hence use your value of $\theta$ to calculate the value of $\beta$. 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_770

In astronomy, the concept of black bodies is very important to better calculate the radiation of stars. Which one is the correct definition of a black body? A: An idealized physical object that reflects all electromagnetic radiation. B: An idealized physical object that absorbs all electromagnetic radiation. C: An idealized physical object that reflects all polarized radiation. D: An idealized physical object that absorbs all polarized radiation.

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: In astronomy, the concept of black bodies is very important to better calculate the radiation of stars. Which one is the correct definition of a black body? A: An idealized physical object that reflects all electromagnetic radiation. B: An idealized physical object that absorbs all electromagnetic radiation. C: An idealized physical object that reflects all polarized radiation. D: An idealized physical object that absorbs all polarized radiation. 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_52

2021 年 5 月, “天问一号”探测器成功在火星软着陆, 我国成为世界上第一个首次探测火星就实现“绕、落、巡”三项任务的国家。 为了简化问题, 可以认为地球和火星在同一平面上绕太阳做匀速圆周运动, 如图 1 所示。已知地球的公转周期为 $T_{1}$, 火星的公转周期为 $T_{2}$ 。 考虑到飞行时间和节省燃料, 地球和火星处于图 1 中相对位置时是在地球上发射火星探测器的最佳时机, 推导在地球上相邻两次发射火星探测器最佳时机的时间间隔 $\Delta t$ 。 [图1] 图1 [图2] 图2

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: 2021 年 5 月, “天问一号”探测器成功在火星软着陆, 我国成为世界上第一个首次探测火星就实现“绕、落、巡”三项任务的国家。 为了简化问题, 可以认为地球和火星在同一平面上绕太阳做匀速圆周运动, 如图 1 所示。已知地球的公转周期为 $T_{1}$, 火星的公转周期为 $T_{2}$ 。 考虑到飞行时间和节省燃料, 地球和火星处于图 1 中相对位置时是在地球上发射火星探测器的最佳时机, 推导在地球上相邻两次发射火星探测器最佳时机的时间间隔 $\Delta t$ 。 [图1] 图1 [图2] 图2 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_195

开普勒发现了行星运动的三大定律, 分别是轨道定律、面积定律和周期定, 这三大定律最 终使他赢得了“天空立法者”的美名, 开普勒第一定律: 所有行星绕太阳运动的轨道都是椭圆, 太阳处在椭圆的一个焦点上. 开普勒第二定律：对任意一个行星来说,它与太阳的连线在相等的时间内扫过相等的面积 开普勒第三定律: 所有行星的轨道的半长轴的三次方跟它的公转周期的二次方的比值都相等即： $\frac{a^{3}}{T^{2}}=K$如图所示, 人造地球卫星在 I 轨道做匀速圆周运动时, 卫星距地面高度为 $h=3 R, R$为地球的半径, 卫星质量为 $m$, 地球表面的重力加速度为 $\mathrm{g}$, 椭圆轨道的长轴 $P Q=10 R$ 。 [图1]求卫星在 I 轨道运动时的速度大小;

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: 开普勒发现了行星运动的三大定律, 分别是轨道定律、面积定律和周期定, 这三大定律最 终使他赢得了“天空立法者”的美名, 开普勒第一定律: 所有行星绕太阳运动的轨道都是椭圆, 太阳处在椭圆的一个焦点上. 开普勒第二定律：对任意一个行星来说,它与太阳的连线在相等的时间内扫过相等的面积 开普勒第三定律: 所有行星的轨道的半长轴的三次方跟它的公转周期的二次方的比值都相等即： $\frac{a^{3}}{T^{2}}=K$如图所示, 人造地球卫星在 I 轨道做匀速圆周运动时, 卫星距地面高度为 $h=3 R, R$为地球的半径, 卫星质量为 $m$, 地球表面的重力加速度为 $\mathrm{g}$, 椭圆轨道的长轴 $P Q=10 R$ 。 [图1]求卫星在 I 轨道运动时的速度大小; 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

multi-modal

Astronomy_935

What is the period of the comet from the previous question? A: 2.1 years B: 2.9 years C: 5.2 years D: 11.2 years

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 period of the comet from the previous question? A: 2.1 years B: 2.9 years C: 5.2 years D: 11.2 years 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_494

我国已掌握“半弹道跳跃式高速再入返回技术”, 为实现“嫦娥”飞船月地返回任务奠定基础。如图所示, 假设与地球同球心的虚线球面为地球大气层边界, 虚线球面外侧没有空气, 返回舱从 $a$ 点无动力滑入大气层, 然后经 $b$ 点从 $c$ 点“跳出”, 再经 $d$ 点从 $e$ 点 “跃入”实现多次减速, 可避免损坏返回舱。 $d$ 点为轨迹最高点, 离地面高 $h$, 已知地球质量为 $M$ ，半径为 $R$ ，引力常量为 $G$ 。则返回舱（） [图1] A: 在 $d$ 点加速度小于 $\frac{G M}{(R+h)^{2}}$ B: 在 $d$ 点速度小于 $\sqrt{\frac{G M}{R+h}}$ C: 虚线球面上 $c 、 e$ 两点离地面高度相等, 所以 $\mathrm{v}_{\mathrm{c}}$ 和 $\mathrm{v}_{\mathrm{e}}$ 大小相等 D: 虚线球面上 $a 、 c$ 两点离地面高度相等, 所以 $\mathrm{v}_{\mathrm{a}}$ 和 $\mathrm{v}_{\mathrm{c}}$ 大小相等

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: 我国已掌握“半弹道跳跃式高速再入返回技术”, 为实现“嫦娥”飞船月地返回任务奠定基础。如图所示, 假设与地球同球心的虚线球面为地球大气层边界, 虚线球面外侧没有空气, 返回舱从 $a$ 点无动力滑入大气层, 然后经 $b$ 点从 $c$ 点“跳出”, 再经 $d$ 点从 $e$ 点 “跃入”实现多次减速, 可避免损坏返回舱。 $d$ 点为轨迹最高点, 离地面高 $h$, 已知地球质量为 $M$ ，半径为 $R$ ，引力常量为 $G$ 。则返回舱（） [图1] A: 在 $d$ 点加速度小于 $\frac{G M}{(R+h)^{2}}$ B: 在 $d$ 点速度小于 $\sqrt{\frac{G M}{R+h}}$ C: 虚线球面上 $c 、 e$ 两点离地面高度相等, 所以 $\mathrm{v}_{\mathrm{c}}$ 和 $\mathrm{v}_{\mathrm{e}}$ 大小相等 D: 虚线球面上 $a 、 c$ 两点离地面高度相等, 所以 $\mathrm{v}_{\mathrm{a}}$ 和 $\mathrm{v}_{\mathrm{c}}$ 大小相等 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_250

已知地球自转周期为 $\mathrm{T}$, 地球半径为 $\mathrm{R}$, 地球极地表面的重力加速度为 $\mathrm{g}$,关于在地球同步轨道运行的卫星, 下列说法正确的是( ) A: 角速度为 $\frac{2 \pi}{T}$ B: 距地高度为 $\sqrt[3]{\frac{g R^{2} T^{2}}{4 \pi^{2}}}$ C: 线速度为 $\sqrt[3]{\frac{2 \pi g R^{2}}{T}}$ D: 加速度为 $\sqrt[3]{\frac{16 g R^{2} \pi^{4}}{T^{4}}}$

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: 已知地球自转周期为 $\mathrm{T}$, 地球半径为 $\mathrm{R}$, 地球极地表面的重力加速度为 $\mathrm{g}$,关于在地球同步轨道运行的卫星, 下列说法正确的是( ) A: 角速度为 $\frac{2 \pi}{T}$ B: 距地高度为 $\sqrt[3]{\frac{g R^{2} T^{2}}{4 \pi^{2}}}$ C: 线速度为 $\sqrt[3]{\frac{2 \pi g R^{2}}{T}}$ D: 加速度为 $\sqrt[3]{\frac{16 g R^{2} \pi^{4}}{T^{4}}}$ 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_1217

In order to move a spacecraft between orbits we must apply a thrust using rockets, which changes the velocity of the spacecraft by $\Delta v$. In this question we will ignore changes in the mass of the spacecraft due to the burning of fuel. For an object of mass $m$ in a circular orbit of radius $r$ around an object with mass $M$ (where $m \ll M$ ) the orbital velocity, $v_{\text {orb }}$, is given by the formula $v_{\text {orb }}=\sqrt{\frac{G M}As part of their plan to rule the galaxy the First Order has created the Starkiller Base. Built within an ice planet and with a superweapon capable of destroying entire star systems, it is charged using the power of stars. The Starkiller Base has moved into the solar system and seeks to use the Sun to power its weapon to destroy the Earth. [figure1] Figure 3: The Starkiller Base charging its superweapon by draining energy from the local star. Credit: Star Wars: The Force Awakens, Lucasfilm. For this question you will need that the gravitational binding energy, $U$, of a uniform density spherical object with mass $M$ and radius $R$ is given by $$ U=\frac{3 G M^{2}}{5 R} $$ and that the mass-luminosity relation of low-mass main sequence stars is given by $L \propto M^{4}$.{r}}$.a. Assume the Sun was initially made of pure hydrogen, carries out nuclear fusion at a constant rate and will continue to do so until the hydrogen in its core is used up. If the mass of the core is $10 \%$ of the star, and $0.7 \%$ of the mass in each fusion reaction is converted into energy, show that the Sun's lifespan on the main sequence is approximately 10 billion 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. Here is some context information for this question, which might assist you in solving it: In order to move a spacecraft between orbits we must apply a thrust using rockets, which changes the velocity of the spacecraft by $\Delta v$. In this question we will ignore changes in the mass of the spacecraft due to the burning of fuel. For an object of mass $m$ in a circular orbit of radius $r$ around an object with mass $M$ (where $m \ll M$ ) the orbital velocity, $v_{\text {orb }}$, is given by the formula $v_{\text {orb }}=\sqrt{\frac{G M}As part of their plan to rule the galaxy the First Order has created the Starkiller Base. Built within an ice planet and with a superweapon capable of destroying entire star systems, it is charged using the power of stars. The Starkiller Base has moved into the solar system and seeks to use the Sun to power its weapon to destroy the Earth. [figure1] Figure 3: The Starkiller Base charging its superweapon by draining energy from the local star. Credit: Star Wars: The Force Awakens, Lucasfilm. For this question you will need that the gravitational binding energy, $U$, of a uniform density spherical object with mass $M$ and radius $R$ is given by $$ U=\frac{3 G M^{2}}{5 R} $$ and that the mass-luminosity relation of low-mass main sequence stars is given by $L \propto M^{4}$.{r}}$. problem: a. Assume the Sun was initially made of pure hydrogen, carries out nuclear fusion at a constant rate and will continue to do so until the hydrogen in its core is used up. If the mass of the core is $10 \%$ of the star, and $0.7 \%$ of the mass in each fusion reaction is converted into energy, show that the Sun's lifespan on the main sequence is approximately 10 billion 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 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_916

Given the mass and radius of an exoplanet we can determine its likely composition, since the denser the material the smaller the planet will be for a given mass (see Figure 3). At a minimum orbital distance, called the Roche limit, tidal forces will cause a planet to break up, so very short period exoplanets place meaningful constraints on their composition as they must be very dense to survive in that orbit. The most extreme variant is called an 'iron planet', as it is made of iron with little or no rocky mantle ( $\gtrsim 80 \%$ iron by mass). The exoplanet KOI 1843.03 has the shortest known orbital period of a little over 4 hours, so is a potential candidate to be an iron planet. [figure1] Figure 3: Left: Comparisons of sizes of planets with different compositions [Marc Kuchner / NASA GSFC]. Right: An artist's impression of what an extreme iron planet (almost $\sim 100 \%$ iron) might look like. The Roche limiting distance, $a_{\min }$, for a body comprised of an incompressible fluid with negligible bulk tensile strength in a circular orbit about its parent star is $$ a_{\min }=2.44 R_{\star}\left(\frac{\rho_{\star}}{\rho_{p}}\right)^{1 / 3} $$ where $R_{\star}$ is the radius of the star, $\rho_{\star}$ is the density of the star and $\rho_{p}$ is the density of the planet. Over the mass range $0.1 M_{E}-1.0 M_{E}$ the mass-radius relation for pure silicate and pure iron planets is approximately a power law and can be described as $$ \log _{10}\left(\frac{R}{R_{E}}\right)=0.295 \log _{10}\left(\frac{M}{M_{E}}\right)+\alpha $$ where $\alpha=0.0286$ in the pure silicate case and $\alpha=-0.1090$ in the pure iron case. The measured period is 0.1768913 days. If it is only made of iron and silicate, estimate the minimum percentage of iron.

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: Given the mass and radius of an exoplanet we can determine its likely composition, since the denser the material the smaller the planet will be for a given mass (see Figure 3). At a minimum orbital distance, called the Roche limit, tidal forces will cause a planet to break up, so very short period exoplanets place meaningful constraints on their composition as they must be very dense to survive in that orbit. The most extreme variant is called an 'iron planet', as it is made of iron with little or no rocky mantle ( $\gtrsim 80 \%$ iron by mass). The exoplanet KOI 1843.03 has the shortest known orbital period of a little over 4 hours, so is a potential candidate to be an iron planet. [figure1] Figure 3: Left: Comparisons of sizes of planets with different compositions [Marc Kuchner / NASA GSFC]. Right: An artist's impression of what an extreme iron planet (almost $\sim 100 \%$ iron) might look like. The Roche limiting distance, $a_{\min }$, for a body comprised of an incompressible fluid with negligible bulk tensile strength in a circular orbit about its parent star is $$ a_{\min }=2.44 R_{\star}\left(\frac{\rho_{\star}}{\rho_{p}}\right)^{1 / 3} $$ where $R_{\star}$ is the radius of the star, $\rho_{\star}$ is the density of the star and $\rho_{p}$ is the density of the planet. Over the mass range $0.1 M_{E}-1.0 M_{E}$ the mass-radius relation for pure silicate and pure iron planets is approximately a power law and can be described as $$ \log _{10}\left(\frac{R}{R_{E}}\right)=0.295 \log _{10}\left(\frac{M}{M_{E}}\right)+\alpha $$ where $\alpha=0.0286$ in the pure silicate case and $\alpha=-0.1090$ in the pure iron case. The measured period is 0.1768913 days. If it is only made of iron and silicate, estimate the minimum percentage of iron. 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 percentage, 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_68

2021 年 5 月 15 日中国首次火星探测任务“天问一号”探测器的着陆巡视器在火星乌托邦平原南部预选着陆区成功着陆, 在火星上首次留下中国印迹, 迈出了中国星际探测征程的重要一步。“天问一号”探测器需要通过霍曼转移轨道从地球发射到火星, 地球轨道和火星轨道近似看成圆形轨道, 霍曼转移轨道是一个在近日点 $M$ 和远日点 $P$ 分别与地球轨道、火星轨道相切的椭圆轨道 (如图所示), 在近日点短暂点火后“天向一号”进入霍曼转移轨道, 接着“天问一号”沿着这个轨道运行直至抵达远日点, 然后再次点火进入火星轨道。已知引力常量为 $G$, 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、“天向一号”运行方向都为逆时针方向。若只考虑太阳对“天问一号”的作用力, 下列说法正确的是（） [图1] A: 两次点火喷射方向都与速度方向相反 B: “天问-号”运行中, 在霍曼转移轨道上 $P$ 点的加速度比在火星轨道上 $P$ 点的加速度小 C: 两次点火之间的时间间隔为 $\frac{\pi}{2 \sqrt{2}} \sqrt{\frac{(R+r)^{3}}{G m}}$ D: “天问一号”在地球轨道上的角速度小于在火星轨道上的角速度

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 2021 年 5 月 15 日中国首次火星探测任务“天问一号”探测器的着陆巡视器在火星乌托邦平原南部预选着陆区成功着陆, 在火星上首次留下中国印迹, 迈出了中国星际探测征程的重要一步。“天问一号”探测器需要通过霍曼转移轨道从地球发射到火星, 地球轨道和火星轨道近似看成圆形轨道, 霍曼转移轨道是一个在近日点 $M$ 和远日点 $P$ 分别与地球轨道、火星轨道相切的椭圆轨道 (如图所示), 在近日点短暂点火后“天向一号”进入霍曼转移轨道, 接着“天问一号”沿着这个轨道运行直至抵达远日点, 然后再次点火进入火星轨道。已知引力常量为 $G$, 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、“天向一号”运行方向都为逆时针方向。若只考虑太阳对“天问一号”的作用力, 下列说法正确的是（） [图1] A: 两次点火喷射方向都与速度方向相反 B: “天问-号”运行中, 在霍曼转移轨道上 $P$ 点的加速度比在火星轨道上 $P$ 点的加速度小 C: 两次点火之间的时间间隔为 $\frac{\pi}{2 \sqrt{2}} \sqrt{\frac{(R+r)^{3}}{G m}}$ D: “天问一号”在地球轨道上的角速度小于在火星轨道上的角速度 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_657

开普勒发现了行星运动的三大定律, 分别是轨道定律、面积定律和周期定, 这三大定律最 终使他赢得了“天空立法者”的美名, 开普勒第一定律: 所有行星绕太阳运动的轨道都是椭圆, 太阳处在椭圆的一个焦点上. 开普勒第二定律：对任意一个行星来说,它与太阳的连线在相等的时间内扫过相等的面积 开普勒第三定律: 所有行星的轨道的半长轴的三次方跟它的公转周期的二次方的比值都相等即： $\frac{a^{3}}{T^{2}}=K$如图所示, 人造地球卫星在 I 轨道做匀速圆周运动时, 卫星距地面高度为 $h=3 R, R$为地球的半径, 卫星质量为 $m$, 地球表面的重力加速度为 $\mathrm{g}$, 椭圆轨道的长轴 $P Q=10 R$ 。 [图1]在牛顿力学体系中, 当两个质量分别为 $m_{1} 、 m_{2}$ 的质点相距为 $r$ 时具有的势能, 称为引力势能, 其大小为 $\mathrm{E}_{\mathrm{P}}=-\frac{G m_{1} m_{2}}{r}$ (规定无穷远处势能为零)卫星在 I 轨道的 $\mathrm{P}$ 点点火加速, 变轨到II轨道,卫星在 I 轨道的 P 点, 变轨到II轨道, 求则至少需对卫星做多少功(不考虑卫星质量的变化和所受的阻力).

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: 开普勒发现了行星运动的三大定律, 分别是轨道定律、面积定律和周期定, 这三大定律最 终使他赢得了“天空立法者”的美名, 开普勒第一定律: 所有行星绕太阳运动的轨道都是椭圆, 太阳处在椭圆的一个焦点上. 开普勒第二定律：对任意一个行星来说,它与太阳的连线在相等的时间内扫过相等的面积 开普勒第三定律: 所有行星的轨道的半长轴的三次方跟它的公转周期的二次方的比值都相等即： $\frac{a^{3}}{T^{2}}=K$如图所示, 人造地球卫星在 I 轨道做匀速圆周运动时, 卫星距地面高度为 $h=3 R, R$为地球的半径, 卫星质量为 $m$, 地球表面的重力加速度为 $\mathrm{g}$, 椭圆轨道的长轴 $P Q=10 R$ 。 [图1]在牛顿力学体系中, 当两个质量分别为 $m_{1} 、 m_{2}$ 的质点相距为 $r$ 时具有的势能, 称为引力势能, 其大小为 $\mathrm{E}_{\mathrm{P}}=-\frac{G m_{1} m_{2}}{r}$ (规定无穷远处势能为零)卫星在 I 轨道的 $\mathrm{P}$ 点点火加速, 变轨到II轨道,卫星在 I 轨道的 P 点, 变轨到II轨道, 求则至少需对卫星做多少功(不考虑卫星质量的变化和所受的阻力). 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_509

春分时，当太阳光直射地球赤道时，某天文爱好者在地球表面上某处用天文望远镜恰好观测到其正上方有一颗被太阳光照射的地球同步卫星。下列关于在日落后的 12 小时内，该观察者看不见此卫星的时间的判断正确的是（已知地球半径为 $R$, 地球表面处的重力加速度为 $g$, 地球自转角速度为 $\omega$, 不考虑大气对光的折射）( ) A: $t=\frac{1}{\omega} \arcsin \sqrt[3]{\frac{R \omega^{2}}{g}}$ B: $t=\frac{2}{\omega} \arcsin \sqrt[3]{\frac{g}{R \omega^{2}}}$ C: $t=\frac{2}{\omega} \arcsin \sqrt[3]{\frac{R \omega^{2}}{g}}$ D: $t=\frac{1}{\omega} \arcsin \sqrt[3]{\frac{g}{R \omega^{2}}}$

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: 春分时，当太阳光直射地球赤道时，某天文爱好者在地球表面上某处用天文望远镜恰好观测到其正上方有一颗被太阳光照射的地球同步卫星。下列关于在日落后的 12 小时内，该观察者看不见此卫星的时间的判断正确的是（已知地球半径为 $R$, 地球表面处的重力加速度为 $g$, 地球自转角速度为 $\omega$, 不考虑大气对光的折射）( ) A: $t=\frac{1}{\omega} \arcsin \sqrt[3]{\frac{R \omega^{2}}{g}}$ B: $t=\frac{2}{\omega} \arcsin \sqrt[3]{\frac{g}{R \omega^{2}}}$ C: $t=\frac{2}{\omega} \arcsin \sqrt[3]{\frac{R \omega^{2}}{g}}$ D: $t=\frac{1}{\omega} \arcsin \sqrt[3]{\frac{g}{R \omega^{2}}}$ 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_401

鸟神星是太阳系内已知的第三大矮行星, 已知其质量为 $m$, 绕太阳做匀速圆周运动(近似认为)的周期为 $T_{1}$, 鸟神星的自转周期为 $T_{2}$, 表面的重力加速度为 $g$, 引力常量为 $G$,根据这些已知量可得 A: 鸟神星的半径为 $\frac{G m}{g}$ B: 鸟神星到太阳的距离为 $\sqrt[3]{\frac{G m T_{1}}{4 \pi}}$ C: 鸟神星的同步卫星的轨道半径为 $\sqrt[3]{\frac{G m T_{2}^{2}}{4 \pi^{2}}}$ D: 鸟神星的第一宇宙速度为 $\sqrt{m g G}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 鸟神星是太阳系内已知的第三大矮行星, 已知其质量为 $m$, 绕太阳做匀速圆周运动(近似认为)的周期为 $T_{1}$, 鸟神星的自转周期为 $T_{2}$, 表面的重力加速度为 $g$, 引力常量为 $G$,根据这些已知量可得 A: 鸟神星的半径为 $\frac{G m}{g}$ B: 鸟神星到太阳的距离为 $\sqrt[3]{\frac{G m T_{1}}{4 \pi}}$ C: 鸟神星的同步卫星的轨道半径为 $\sqrt[3]{\frac{G m T_{2}^{2}}{4 \pi^{2}}}$ D: 鸟神星的第一宇宙速度为 $\sqrt{m g G}$ 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应为以下选项之一：[A, B, C, D]

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Astronomy

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text-only

Astronomy_781

Why are Cepheid stars relevant for astronomers? A: To measure interstellar mass. B: To measure galactic distances. C: To measure galactic energy-density. D: To measure interstellar density.

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 are Cepheid stars relevant for astronomers? A: To measure interstellar mass. B: To measure galactic distances. C: To measure galactic energy-density. D: To measure interstellar density. 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_413

2018 年 6 月 14 日。承担嫦娥四号中继通信任务的“鹊桥”中继星抵达绕地月第二拉 格朗日点的轨道, 第二拉格朗日点是地月连线延长线上的一点, 处于该位置上的卫星与月球同步绕地球公转, 则该卫星的（） [图1] A: 向心力仅来自于地球引力 B: 线速度大于月球的线速度 C: 角速度大于月球的角速度 D: 向心加速度大于月球的向心加速度

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 2018 年 6 月 14 日。承担嫦娥四号中继通信任务的“鹊桥”中继星抵达绕地月第二拉 格朗日点的轨道, 第二拉格朗日点是地月连线延长线上的一点, 处于该位置上的卫星与月球同步绕地球公转, 则该卫星的（） [图1] A: 向心力仅来自于地球引力 B: 线速度大于月球的线速度 C: 角速度大于月球的角速度 D: 向心加速度大于月球的向心加速度 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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multi-modal

Astronomy_884

A student group from California Institute of Technology is planning a massive prank for this year. This year, they are planning on launching a rocket to Massachusetts Institute of Technology and making it explode above the so called 'Great Dome' of MIT. The rocket will contain a giant parachute printed with a logo of Caltech, so that the Great Dome will be covered with Caltech logo. How much distance does this rocket need to fly? The longitude and latitude information of two locations are given in the table below. (Assume that the Earth is a perfect sphere with radius of $6371 \mathrm{~km}$.) | | Latitude | Longtitude | | :--- | :--- | :--- | | The Great Dome of <br> MIT | $42.3601^{\circ} \mathrm{N}$ | $71.0942^{\circ} \mathrm{W}$ | | Caltech | $34.1377^{\circ} \mathrm{N}$ | $118.1253^{\circ} \mathrm{W} A: $3890 \mathrm{~km}$ B: $4160 \mathrm{~km}$ C: $4780 \mathrm{~km}$ D: $4910 \mathrm{~km}$ E: $5290 \mathrm{~km}$

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: A student group from California Institute of Technology is planning a massive prank for this year. This year, they are planning on launching a rocket to Massachusetts Institute of Technology and making it explode above the so called 'Great Dome' of MIT. The rocket will contain a giant parachute printed with a logo of Caltech, so that the Great Dome will be covered with Caltech logo. How much distance does this rocket need to fly? The longitude and latitude information of two locations are given in the table below. (Assume that the Earth is a perfect sphere with radius of $6371 \mathrm{~km}$.) | | Latitude | Longtitude | | :--- | :--- | :--- | | The Great Dome of <br> MIT | $42.3601^{\circ} \mathrm{N}$ | $71.0942^{\circ} \mathrm{W}$ | | Caltech | $34.1377^{\circ} \mathrm{N}$ | $118.1253^{\circ} \mathrm{W} A: $3890 \mathrm{~km}$ B: $4160 \mathrm{~km}$ C: $4780 \mathrm{~km}$ D: $4910 \mathrm{~km}$ E: $5290 \mathrm{~km}$ 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, E].

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Astronomy_148

如图所示, 从地面上 $A$ 点发射一枚远程弹道导弹, 假设导弹仅在地球引力作用下沿 $A C B$ 粗圆轨道飞行并击中地面目标 $B, C$ 为轨道的远地点, 距地面高度为 $h$. 已知地球半径为 $R$, 地球质量为 $M$, 引力常量为 $G$. 则下列结论正确的是 ( ) [图1] A: 导弹在 $C$ 点的速度大于 $\sqrt{\frac{G M}{R+h}}$ B: 导弹在 $C$ 点的速度等于 $\sqrt[3]{\frac{G M}{R+h}}$ C: 导弹在 $c$ 点的加速度等于 $\frac{G M}{(R+h)^{2}}$ D: 导弹在 $C$ 点的加速度大于 $\frac{G M}{(R-h)^{2}}$

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, 从地面上 $A$ 点发射一枚远程弹道导弹, 假设导弹仅在地球引力作用下沿 $A C B$ 粗圆轨道飞行并击中地面目标 $B, C$ 为轨道的远地点, 距地面高度为 $h$. 已知地球半径为 $R$, 地球质量为 $M$, 引力常量为 $G$. 则下列结论正确的是 ( ) [图1] A: 导弹在 $C$ 点的速度大于 $\sqrt{\frac{G M}{R+h}}$ B: 导弹在 $C$ 点的速度等于 $\sqrt[3]{\frac{G M}{R+h}}$ C: 导弹在 $c$ 点的加速度等于 $\frac{G M}{(R+h)^{2}}$ D: 导弹在 $C$ 点的加速度大于 $\frac{G M}{(R-h)^{2}}$ 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_474

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”号和气囊第一次与火星碰撞时所受到的平均冲力。

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”号和气囊第一次与火星碰撞时所受到的平均冲力。 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N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_861

An interesting phenomena that happens in the Solar System is the capture of comets in the interstellar medium. Assume that a comet with a mass of $7.15 * 10^{16} \mathrm{~kg}$ is captured by the solar system. The perihelion of this comet's orbit after it is captured is equal to $4.64 \mathrm{AU}$, and its velocity with respect to the Sun before being captured by the Solar System was very small. Calculate the velocity of the comet at the perihelion. A: $87.1 \mathrm{~km} / \mathrm{s}$ B: $45.9 \mathrm{~km} / \mathrm{s}$ C: $5.67 \mathrm{~km} / \mathrm{s}$ D: $105.4 \mathrm{~km} / \mathrm{s}$ E: $19.6 \mathrm{~km} / \mathrm{s}$

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: An interesting phenomena that happens in the Solar System is the capture of comets in the interstellar medium. Assume that a comet with a mass of $7.15 * 10^{16} \mathrm{~kg}$ is captured by the solar system. The perihelion of this comet's orbit after it is captured is equal to $4.64 \mathrm{AU}$, and its velocity with respect to the Sun before being captured by the Solar System was very small. Calculate the velocity of the comet at the perihelion. A: $87.1 \mathrm{~km} / \mathrm{s}$ B: $45.9 \mathrm{~km} / \mathrm{s}$ C: $5.67 \mathrm{~km} / \mathrm{s}$ D: $105.4 \mathrm{~km} / \mathrm{s}$ E: $19.6 \mathrm{~km} / \mathrm{s}$ 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_410

如图所示, 在某行星表面上有一倾斜的圆盘, 面与水平面的夹角为 $30^{\circ}$, 盘面上离转轴距离 $L$ 处有小物体与圆盘保持相对静止, 绕垂直于盘面的固定对称轴以恒定角速度转动, 角速度为 $\omega$ 时, 小物块刚要滑动, 物体与盘面间的动摩擦因数为 $\frac{2 \sqrt{3}}{3}$ (设最大静摩擦力等于滑动摩擦力), 星球的半径为 $R$, 引力常量为 $G$, 下列说法正确的是 ( ) [图1] A: 这个行星的质量 $\frac{2 \omega^{2} R^{2} L}{G}$ B: 这个行星的第一宇宙速度 $v=2 \omega \sqrt{L R}$ C: 这个行星的密度是 $\rho=\frac{3 \omega^{2} L}{\pi G R}$ D: 离行星表面距离为 $R$ 的地方的重力加速度为 $\frac{1}{2} \omega^{2} L$

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, 在某行星表面上有一倾斜的圆盘, 面与水平面的夹角为 $30^{\circ}$, 盘面上离转轴距离 $L$ 处有小物体与圆盘保持相对静止, 绕垂直于盘面的固定对称轴以恒定角速度转动, 角速度为 $\omega$ 时, 小物块刚要滑动, 物体与盘面间的动摩擦因数为 $\frac{2 \sqrt{3}}{3}$ (设最大静摩擦力等于滑动摩擦力), 星球的半径为 $R$, 引力常量为 $G$, 下列说法正确的是 ( ) [图1] A: 这个行星的质量 $\frac{2 \omega^{2} R^{2} L}{G}$ B: 这个行星的第一宇宙速度 $v=2 \omega \sqrt{L R}$ C: 这个行星的密度是 $\rho=\frac{3 \omega^{2} L}{\pi G R}$ D: 离行星表面距离为 $R$ 的地方的重力加速度为 $\frac{1}{2} \omega^{2} L$ 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_343

13. 一同学准备设计一个绕地球纬度圈飞行的卫星, 绕行方向与地球自转方向相同, 且要求其在一天绕地球 3 周,则该卫星与地球静止同步卫星相比,下列说法正确的是 ( ) A: 该卫星与地球静止同步卫星可能不在同一轨道平面内 B: 该卫星离地高度与地球静止同步卫星的离地高度之比为 $\left(\frac{1}{9}\right)^{\frac{1}{3}}$ C: 该卫星线速度与地球静止同步卫星的线速度之比为 $4^{\frac{1}{6}}$ D: 该卫星与地球静止同步卫星的向心加速度之比为 $3^{\frac{4}{3}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 13. 一同学准备设计一个绕地球纬度圈飞行的卫星, 绕行方向与地球自转方向相同, 且要求其在一天绕地球 3 周,则该卫星与地球静止同步卫星相比,下列说法正确的是 ( ) A: 该卫星与地球静止同步卫星可能不在同一轨道平面内 B: 该卫星离地高度与地球静止同步卫星的离地高度之比为 $\left(\frac{1}{9}\right)^{\frac{1}{3}}$ C: 该卫星线速度与地球静止同步卫星的线速度之比为 $4^{\frac{1}{6}}$ D: 该卫星与地球静止同步卫星的向心加速度之比为 $3^{\frac{4}{3}}$ 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应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

text-only

Astronomy_122

有一轨道平面与赤道平面重合的侦察卫星, 轨道高度为 $R$, 飞行方向与地球自转方向相同。设地球自转周期为 $T_{0}$, 半径为 $R$, 地球赤道处的重力加速度为 $g$ 。位于赤道的某一地面基站在某时刻恰好与该卫星建立起通信链路, 则该地面基站能不间断的从侦察卫星上下载侦察数据的时间为（） A: A. $\frac{2 \pi}{3\left(\sqrt{\frac{g}{R}}-\frac{2 \pi}{T_{0}}\right)}$ B: [图1] C: [图2] D: [图3]

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 有一轨道平面与赤道平面重合的侦察卫星, 轨道高度为 $R$, 飞行方向与地球自转方向相同。设地球自转周期为 $T_{0}$, 半径为 $R$, 地球赤道处的重力加速度为 $g$ 。位于赤道的某一地面基站在某时刻恰好与该卫星建立起通信链路, 则该地面基站能不间断的从侦察卫星上下载侦察数据的时间为（） A: A. $\frac{2 \pi}{3\left(\sqrt{\frac{g}{R}}-\frac{2 \pi}{T_{0}}\right)}$ B: [图1] C: [图2] D: [图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应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

multi-modal

Astronomy_1205

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 $q=0.251 \rho X^{2} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \mathrm{~W} \mathrm{~kg}^{-1}$.b. The Sun is composed predominantly of ionized hydrogen and helium, with approximate mass fractions $X=0.70$ and $Y=0.30$ respectively (taken to be constant throughout the Sun). ii. Using the Virial Theorem, and given $E_{G} \approx G M_{\odot}^{2} / R_{\odot}$, estimate the Sun's mean temperature.

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: b. The Sun is composed predominantly of ionized hydrogen and helium, with approximate mass fractions $X=0.70$ and $Y=0.30$ respectively (taken to be constant throughout the Sun). ii. Using the Virial Theorem, and given $E_{G} \approx G M_{\odot}^{2} / R_{\odot}$, estimate the Sun's mean temperature. 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_157

如图所示, 探月卫星的发射过程可简化如下：首先进入绕地球运行的“停泊轨道”,在该轨道的 $P$ 处通过变速再进入“地月转移轨道”, 在快要到达月球时, 对卫星再次变速,卫星被月球引力“俘获”后, 成为环月卫星, 最终在环绕月球的“工作轨道”绕月飞行（视为圆周运动), 对月球进行探测. “工作轨道”周期为 $T$ 、距月球表面的高度为 $h$, 月球半径为 $R$, 引力常量为 $G$, 忽略其他天体对探月卫星在“工作轨道”上环绕运动的影响。 求探月卫星在“工作轨道”上环绕的线速度大小; [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: 如图所示, 探月卫星的发射过程可简化如下：首先进入绕地球运行的“停泊轨道”,在该轨道的 $P$ 处通过变速再进入“地月转移轨道”, 在快要到达月球时, 对卫星再次变速,卫星被月球引力“俘获”后, 成为环月卫星, 最终在环绕月球的“工作轨道”绕月飞行（视为圆周运动), 对月球进行探测. “工作轨道”周期为 $T$ 、距月球表面的高度为 $h$, 月球半径为 $R$, 引力常量为 $G$, 忽略其他天体对探月卫星在“工作轨道”上环绕运动的影响。 求探月卫星在“工作轨道”上环绕的线速度大小; [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_1126

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 $$ \Delta l=\int_{r_{2}}^{r_{1}}\left(1-\frac{2 r_{g}}{r}\right)^{-1 / 2} \mathrm{~d} r $$e. Taking the mass of $\mathrm{M}_{8} 7^{*}$ as $6.5 \times 10^{9} \mathrm{M}_{\odot}$ : iii. Determine the minimum and maximum ISCO periods for Sgr A* and hence suggest a possible reason why $\mathrm{M} 87$ has been imaged first, even though Sgr A* has a larger angular diameter, given that each 'exposure' with the EHT was 7 mins long (with multiple exposures from each observing run added together for the final image from each night).

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 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: e. Taking the mass of $\mathrm{M}_{8} 7^{*}$ as $6.5 \times 10^{9} \mathrm{M}_{\odot}$ : iii. Determine the minimum and maximum ISCO periods for Sgr A* and hence suggest a possible reason why $\mathrm{M} 87$ has been imaged first, even though Sgr A* has a larger angular diameter, given that each 'exposure' with the EHT was 7 mins long (with multiple exposures from each observing run added together for the final image from each night). 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{~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_446

在星球 $\mathrm{M}$ 上一轻弹簧坚直固定于水平桌面, 物体 $P$ 轻放在弹簧上由静止释放, 其加速度 $a$ 与弹簧压缩量 $x$ 的关系如图 $P$ 线所示。另一星球 $\mathrm{N}$ 上用完全相同的弹簧, 改用物体 $Q$ 完成同样过程, 其加速度 $a$ 与弹簧压缩量 $x$ 的关系如 $Q$ 线所示, 下列说法正确的是 ( ) [图1] A: 同一物体在 $M$ 星球表面与在 $N$ 星球表面重力大小之比为 3: 1 B: 物体 $P 、 Q$ 的质量之比是 $6: 1$ C: $\mathrm{M}$ 星球上物体 $\mathrm{R}$ 由静止开始做加速度为 $3 a_{0}$ 的匀加速直线运动, 通过位移 $x_{0}$ 时的速度为 $\sqrt{3 a_{0} x_{0}}$ D: 图中 $P 、 Q$ 下落的最大速度之比为 $\frac{\sqrt{6}}{2}$

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: 在星球 $\mathrm{M}$ 上一轻弹簧坚直固定于水平桌面, 物体 $P$ 轻放在弹簧上由静止释放, 其加速度 $a$ 与弹簧压缩量 $x$ 的关系如图 $P$ 线所示。另一星球 $\mathrm{N}$ 上用完全相同的弹簧, 改用物体 $Q$ 完成同样过程, 其加速度 $a$ 与弹簧压缩量 $x$ 的关系如 $Q$ 线所示, 下列说法正确的是 ( ) [图1] A: 同一物体在 $M$ 星球表面与在 $N$ 星球表面重力大小之比为 3: 1 B: 物体 $P 、 Q$ 的质量之比是 $6: 1$ C: $\mathrm{M}$ 星球上物体 $\mathrm{R}$ 由静止开始做加速度为 $3 a_{0}$ 的匀加速直线运动, 通过位移 $x_{0}$ 时的速度为 $\sqrt{3 a_{0} x_{0}}$ D: 图中 $P 、 Q$ 下落的最大速度之比为 $\frac{\sqrt{6}}{2}$ 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_908

On the evening of 27th July 2018, there was a total lunar eclipse (where the Moon passes into the Earth's shadow). As viewed from Oxford, totality began whilst the Moon was below the horizon. Interestingly, moonrise was at $8: 55 \mathrm{pm}$, yet sunset was only at $9: 01 \mathrm{pm}$, so for 6 minutes both the fully eclipsed Moon and setting Sun were visible above the horizon. This very rare event is known as a selenelion. What is the explanation behind this seemingly impossible observation? A: The Moon is in an orbit with a non-zero eccentricity B: Prominences on the Sun at the time of the eclipse C: The non-zero light travel time to cover the large distance between the Moon and the Earth D: The effect of atmospheric refraction meaning the Sun and Moon only appear to be above the horizon

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: On the evening of 27th July 2018, there was a total lunar eclipse (where the Moon passes into the Earth's shadow). As viewed from Oxford, totality began whilst the Moon was below the horizon. Interestingly, moonrise was at $8: 55 \mathrm{pm}$, yet sunset was only at $9: 01 \mathrm{pm}$, so for 6 minutes both the fully eclipsed Moon and setting Sun were visible above the horizon. This very rare event is known as a selenelion. What is the explanation behind this seemingly impossible observation? A: The Moon is in an orbit with a non-zero eccentricity B: Prominences on the Sun at the time of the eclipse C: The non-zero light travel time to cover the large distance between the Moon and the Earth D: The effect of atmospheric refraction meaning the Sun and Moon only appear to be above the horizon 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_90

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 $\mathrm{P}$ 轻放在弹簧上端, 如图 (a) 所示, $\mathrm{P}$ 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 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。 若将星球 $\mathrm{A}$ 看成是以星球 $\mathrm{B}$ 为中心天体的一颗卫星, 星球 $\mathrm{A}$ 的运行周期为 $T_{1}$ 若将星球 $\mathrm{A}$ 和星球 $\mathrm{B}$ 看成是远离其他星球的双星模型, 这样算得的两星球做匀速圆周运动的周期为 $T_{2}$ 。求此情形中的周期 $T_{2}$ 与上述周期 $T_{1}$ 的比值。 [图1] 图(a) [图2] 图(b)

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: 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 $\mathrm{P}$ 轻放在弹簧上端, 如图 (a) 所示, $\mathrm{P}$ 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 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。 若将星球 $\mathrm{A}$ 看成是以星球 $\mathrm{B}$ 为中心天体的一颗卫星, 星球 $\mathrm{A}$ 的运行周期为 $T_{1}$ 若将星球 $\mathrm{A}$ 和星球 $\mathrm{B}$ 看成是远离其他星球的双星模型, 这样算得的两星球做匀速圆周运动的周期为 $T_{2}$ 。求此情形中的周期 $T_{2}$ 与上述周期 $T_{1}$ 的比值。 [图1] 图(a) [图2] 图(b) 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

multi-modal

Astronomy_673

太阳系各行星几乎在同一平面内沿同一方向绕太阳做圆周运动。当地球恰好运行到某地外行星和太阳之间, 且三者几乎排成一条直线的现象, 天文学称为“行星冲日”。已知地球及各地外行星绕太阳运动的轨道半径如表所示（） | | 地球 | 火星 | 木星 | 土星 | 天王星 | 海王星 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | 轨道半径 <br> R/AU | 1.0 | 1.5 | 5.2 | 9.5 | 19 | 30 | A: 土星相邻两次冲日的时间间隔是 450 天左右 B: 土星相邻两次冲日的时间间隔是 378 天左右 C: 火星相邻两次冲日的时间间隔最长 D: 海王星相邻两次冲日的时间间隔最长

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: 太阳系各行星几乎在同一平面内沿同一方向绕太阳做圆周运动。当地球恰好运行到某地外行星和太阳之间, 且三者几乎排成一条直线的现象, 天文学称为“行星冲日”。已知地球及各地外行星绕太阳运动的轨道半径如表所示（） | | 地球 | 火星 | 木星 | 土星 | 天王星 | 海王星 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | 轨道半径 <br> R/AU | 1.0 | 1.5 | 5.2 | 9.5 | 19 | 30 | A: 土星相邻两次冲日的时间间隔是 450 天左右 B: 土星相邻两次冲日的时间间隔是 378 天左右 C: 火星相邻两次冲日的时间间隔最长 D: 海王星相邻两次冲日的时间间隔最长 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_180

开普勒发现了行星运动的三大定律, 分别是轨道定律、面积定律和周期定, 这三大定律最 终使他赢得了“天空立法者”的美名, 开普勒第一定律: 所有行星绕太阳运动的轨道都是椭圆, 太阳处在椭圆的一个焦点上. 开普勒第二定律：对任意一个行星来说,它与太阳的连线在相等的时间内扫过相等的面积 开普勒第三定律: 所有行星的轨道的半长轴的三次方跟它的公转周期的二次方的比值都相等即： $\frac{a^{3}}{T^{2}}=K$如图所示, 人造地球卫星在 I 轨道做匀速圆周运动时, 卫星距地面高度为 $h=3 R, R$为地球的半径, 卫星质量为 $m$, 地球表面的重力加速度为 $\mathrm{g}$, 椭圆轨道的长轴 $P Q=10 R$ 。 [图1]在牛顿力学体系中, 当两个质量分别为 $m_{1} 、 m_{2}$ 的质点相距为 $r$ 时具有的势能, 称为引力势能, 其大小为 $\mathrm{E}_{\mathrm{P}}=-\frac{G m_{1} m_{2}}{r}$ (规定无穷远处势能为零)卫星在 I 轨道的 $\mathrm{P}$ 点点火加速, 变轨到II轨道,根据开普勒第二定律, 求卫星在椭圆轨道II运动时, 在近地点 $\mathrm{P}$ 与在远地点 $\mathrm{Q}$ 的速率之比

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: 开普勒发现了行星运动的三大定律, 分别是轨道定律、面积定律和周期定, 这三大定律最 终使他赢得了“天空立法者”的美名, 开普勒第一定律: 所有行星绕太阳运动的轨道都是椭圆, 太阳处在椭圆的一个焦点上. 开普勒第二定律：对任意一个行星来说,它与太阳的连线在相等的时间内扫过相等的面积 开普勒第三定律: 所有行星的轨道的半长轴的三次方跟它的公转周期的二次方的比值都相等即： $\frac{a^{3}}{T^{2}}=K$如图所示, 人造地球卫星在 I 轨道做匀速圆周运动时, 卫星距地面高度为 $h=3 R, R$为地球的半径, 卫星质量为 $m$, 地球表面的重力加速度为 $\mathrm{g}$, 椭圆轨道的长轴 $P Q=10 R$ 。 [图1]在牛顿力学体系中, 当两个质量分别为 $m_{1} 、 m_{2}$ 的质点相距为 $r$ 时具有的势能, 称为引力势能, 其大小为 $\mathrm{E}_{\mathrm{P}}=-\frac{G m_{1} m_{2}}{r}$ (规定无穷远处势能为零)卫星在 I 轨道的 $\mathrm{P}$ 点点火加速, 变轨到II轨道,根据开普勒第二定律, 求卫星在椭圆轨道II运动时, 在近地点 $\mathrm{P}$ 与在远地点 $\mathrm{Q}$ 的速率之比 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

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Astronomy_533

如图所示, 探月卫星的发射过程可简化如下：首先进入绕地球运行的“停泊轨道”,在该轨道的 $P$ 处通过变速再进入“地月转移轨道”, 在快要到达月球时, 对卫星再次变速,卫星被月球引力“俘获”后, 成为环月卫星, 最终在环绕月球的“工作轨道”绕月飞行（视为圆周运动), 对月球进行探测. “工作轨道”周期为 $T$ 、距月球表面的高度为 $h$, 月球半径为 $R$, 引力常量为 $G$, 忽略其他天体对探月卫星在“工作轨道”上环绕运动的影响。 求月球的第一宇宙速度。 [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, 探月卫星的发射过程可简化如下：首先进入绕地球运行的“停泊轨道”,在该轨道的 $P$ 处通过变速再进入“地月转移轨道”, 在快要到达月球时, 对卫星再次变速,卫星被月球引力“俘获”后, 成为环月卫星, 最终在环绕月球的“工作轨道”绕月飞行（视为圆周运动), 对月球进行探测. “工作轨道”周期为 $T$ 、距月球表面的高度为 $h$, 月球半径为 $R$, 引力常量为 $G$, 忽略其他天体对探月卫星在“工作轨道”上环绕运动的影响。 求月球的第一宇宙速度。 [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

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Astronomy_404

2021 年 1 月 20 日, 我国在西昌卫星发射中心用长征三号乙运载火箭, 成功将天通一号 03 星发射升空, 它将与天通一号 01 星、 02 星组网运行。若 03 星绕地球做圆周运动的轨道半径为 02 星的 $a$ 倍, 02 星做圆周运动的向心加速度为 01 星的 $b$ 倍, 已知 01 星的运行周期为 $T$ ，则 03 星的运行周期为 A: $a^{\frac{3}{2}} b^{\frac{2}{3}} T$ B: $a^{\frac{3}{2}} b^{-\frac{1}{2}} T$ C: $a^{\frac{3}{2}} b^{-\frac{3}{4}} T$ D: $a^{\frac{3}{2}} b^{\frac{3}{4}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 2021 年 1 月 20 日, 我国在西昌卫星发射中心用长征三号乙运载火箭, 成功将天通一号 03 星发射升空, 它将与天通一号 01 星、 02 星组网运行。若 03 星绕地球做圆周运动的轨道半径为 02 星的 $a$ 倍, 02 星做圆周运动的向心加速度为 01 星的 $b$ 倍, 已知 01 星的运行周期为 $T$ ，则 03 星的运行周期为 A: $a^{\frac{3}{2}} b^{\frac{2}{3}} T$ B: $a^{\frac{3}{2}} b^{-\frac{1}{2}} T$ C: $a^{\frac{3}{2}} b^{-\frac{3}{4}} T$ D: $a^{\frac{3}{2}} b^{\frac{3}{4}}$ 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应为以下选项之一：[A, B, C, D]

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Astronomy

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Astronomy_744

The famous Drake equation attempts to answer the following question: A: Will the Sun become a black hole? B: Is the universe infinitely large? C: How old is the visible universe? D: Are we alone in the universe?

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 famous Drake equation attempts to answer the following question: A: Will the Sun become a black hole? B: Is the universe infinitely large? C: How old is the visible universe? D: Are we alone in the universe? 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_904

A comet is orbiting the Sun in an orbit with a semi-major axis of 5.0 au and an eccentricity, $e=0.80$. Calculate its semi-minor axis, $b$. A: $1.0 \mathrm{au}$ B: $2.0 \mathrm{au}$ C: $3.0 \mathrm{au}$ D: $4.0 \mathrm{au}$

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 is orbiting the Sun in an orbit with a semi-major axis of 5.0 au and an eccentricity, $e=0.80$. Calculate its semi-minor axis, $b$. A: $1.0 \mathrm{au}$ B: $2.0 \mathrm{au}$ C: $3.0 \mathrm{au}$ D: $4.0 \mathrm{au}$ 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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Astronomy_788

Which one of the following expressions is correct for the Lorentz factor $\gamma$ and $\beta=$ $v / c$ ? A: $\frac{1}{\gamma^{2}}=1+\beta^{2}$ B: $\frac{1}{\gamma^{2}}=\beta \gamma^{2}$ C: $\frac{1}{\gamma^{2}}=1-\beta^{2}$

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: Which one of the following expressions is correct for the Lorentz factor $\gamma$ and $\beta=$ $v / c$ ? A: $\frac{1}{\gamma^{2}}=1+\beta^{2}$ B: $\frac{1}{\gamma^{2}}=\beta \gamma^{2}$ C: $\frac{1}{\gamma^{2}}=1-\beta^{2}$ 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].

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Astronomy

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Astronomy_794

At which of the following phases of the Moon's orbit is the tidal bulge of Earth largest? A: Full B: First Quarter C: Waxing Gibbous D: Waning Gibbous E: Waxing Crescent

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: At which of the following phases of the Moon's orbit is the tidal bulge of Earth largest? A: Full B: First Quarter C: Waxing Gibbous D: Waning Gibbous E: Waxing Crescent 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].

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Astronomy

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Astronomy_119

如图所示, I为北斗卫星导航系统中的静止轨道卫星, 其对地张角为 $2 \theta$; II为地球试卷第 75 页，共 122 页 的近地卫星。已知地球的自转周期为 $T_{0}$, 万有引力常量为 $G$, 根据题中条件, 可求出 [图1] A: 地球的平均密度为 $\frac{3 \pi}{G T_{0}^{2} \sin ^{3} \theta}$ B: 卫星I和卫星II的加速度之比为 $\sin ^{2} 2 \theta$ C: 卫星II的周期为 $\frac{T_{0}}{\sqrt{\sin ^{3} \theta}}$ D: 卫星II运动的周期内无法直接接收到卫星发出电磁波信号的时间为 $\frac{(\pi+2 \theta) T_{0}}{2 \pi} \sqrt{\sin ^{3} \theta}$

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, I为北斗卫星导航系统中的静止轨道卫星, 其对地张角为 $2 \theta$; II为地球试卷第 75 页，共 122 页 的近地卫星。已知地球的自转周期为 $T_{0}$, 万有引力常量为 $G$, 根据题中条件, 可求出 [图1] A: 地球的平均密度为 $\frac{3 \pi}{G T_{0}^{2} \sin ^{3} \theta}$ B: 卫星I和卫星II的加速度之比为 $\sin ^{2} 2 \theta$ C: 卫星II的周期为 $\frac{T_{0}}{\sqrt{\sin ^{3} \theta}}$ D: 卫星II运动的周期内无法直接接收到卫星发出电磁波信号的时间为 $\frac{(\pi+2 \theta) T_{0}}{2 \pi} \sqrt{\sin ^{3} \theta}$ 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