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GAIR 62

Astronomy_471

如图所示, 两星球相距为 $L$, 质量比为 $\mathrm{m}_{\mathrm{A}}: \mathrm{m}_{\mathrm{B}}=1: 9$, 两星球半径远小于 $L$. 从星球 $A$ 沿 $A 、 B$ 连线向 $B$ 以某一初速度发射一探测器. 只考虑星球 $A 、 B$ 对探测器的作用,下列说法正确的是( ) [图1] A: 探测器的速度一直减小 B: 探测器在距星球 $A$ 为 $\frac{L}{4}$ 处加速度为零 C: 若探测器能到达星球 $B$, 其速度可能恰好为零 D: 若探测器能到达星球 $B$, 其速度一定大于发射时的初速度

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: 如图所示, 两星球相距为 $L$, 质量比为 $\mathrm{m}_{\mathrm{A}}: \mathrm{m}_{\mathrm{B}}=1: 9$, 两星球半径远小于 $L$. 从星球 $A$ 沿 $A 、 B$ 连线向 $B$ 以某一初速度发射一探测器. 只考虑星球 $A 、 B$ 对探测器的作用,下列说法正确的是( ) [图1] A: 探测器的速度一直减小 B: 探测器在距星球 $A$ 为 $\frac{L}{4}$ 处加速度为零 C: 若探测器能到达星球 $B$, 其速度可能恰好为零 D: 若探测器能到达星球 $B$, 其速度一定大于发射时的初速度 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]

MC

Astronomy

ZH

multi-modal

Astronomy_1157

Recently a group of researchers announced that they had discovered an Earth-sized exoplanet around our nearest star, Proxima Centauri. Its closeness raises an intriguing possibility about whether or not we might be able to image it directly using telescopes. The difficulty comes from the small angular scales that need to be resolved and the extreme differences in brightness between the reflected light from the planet and the light given out by the star. [figure1] Figure 6: Artist's impression of the view from the surface of Proxima Centauri b. Credit: ESO / M. Kornmesser Data about the star and the planet are summarised below: | Proxima Centauri (star) | | Proxima Centauri b (planet) | | | :--- | :--- | :--- | :--- | | Distance | $1.295 \mathrm{pc}$ | Orbital period | 11.186 days | | Mass | $0.123 \mathrm{M}_{\odot}$ | Mass $(\mathrm{min})$ | $\approx 1.27 \mathrm{M}_{\oplus}$ | | Radius | $0.141 \mathrm{R}_{\odot}$ | Radius $(\mathrm{min})$ | $\approx 1.1 \mathrm{R}_{\oplus}$ | | Surface temperature | $3042 \mathrm{~K}$ | | | | Apparent magnitude | 11.13 | | | The following formulae may also be helpful: $$ m-\mathcal{M}=5 \log \left(\frac{d}{10}\right) \quad \mathcal{M}-\mathcal{M}_{\odot}=-2.5 \log \left(\frac{L}{\mathrm{~L}_{\odot}}\right) \quad \Delta m=2.5 \log C R $$ where $m$ is the apparent magnitude, $\mathcal{M}$ is the absolute magnitude, $d$ is the distance in parsecs, and the contrast ratio $(C R)$ is defined as the ratio of fluxes from the star and planet, $C R=\frac{f_{\text {star }}}{f_{\text {planet }}}$.a. Calculate the maximum angular separation between the star and the planet, assuming a circular orbit. Give your answer in arcseconds (where $3600 \operatorname{arcseconds}=1^{\circ}$ ).

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: Recently a group of researchers announced that they had discovered an Earth-sized exoplanet around our nearest star, Proxima Centauri. Its closeness raises an intriguing possibility about whether or not we might be able to image it directly using telescopes. The difficulty comes from the small angular scales that need to be resolved and the extreme differences in brightness between the reflected light from the planet and the light given out by the star. [figure1] Figure 6: Artist's impression of the view from the surface of Proxima Centauri b. Credit: ESO / M. Kornmesser Data about the star and the planet are summarised below: | Proxima Centauri (star) | | Proxima Centauri b (planet) | | | :--- | :--- | :--- | :--- | | Distance | $1.295 \mathrm{pc}$ | Orbital period | 11.186 days | | Mass | $0.123 \mathrm{M}_{\odot}$ | Mass $(\mathrm{min})$ | $\approx 1.27 \mathrm{M}_{\oplus}$ | | Radius | $0.141 \mathrm{R}_{\odot}$ | Radius $(\mathrm{min})$ | $\approx 1.1 \mathrm{R}_{\oplus}$ | | Surface temperature | $3042 \mathrm{~K}$ | | | | Apparent magnitude | 11.13 | | | The following formulae may also be helpful: $$ m-\mathcal{M}=5 \log \left(\frac{d}{10}\right) \quad \mathcal{M}-\mathcal{M}_{\odot}=-2.5 \log \left(\frac{L}{\mathrm{~L}_{\odot}}\right) \quad \Delta m=2.5 \log C R $$ where $m$ is the apparent magnitude, $\mathcal{M}$ is the absolute magnitude, $d$ is the distance in parsecs, and the contrast ratio $(C R)$ is defined as the ratio of fluxes from the star and planet, $C R=\frac{f_{\text {star }}}{f_{\text {planet }}}$. problem: a. Calculate the maximum angular separation between the star and the planet, assuming a circular orbit. Give your answer in arcseconds (where $3600 \operatorname{arcseconds}=1^{\circ}$ ). 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_338

已知某卫星在赤道上空轨道半径为 $r_{1}$ 的圆形轨道上绕地球运行的周期为 $T$, 卫星运动方向与地球自转方向相同, 赤道上某城市的人每三天恰好五次看到该卫星掠过其正上方。假设某时刻该卫星在 $A$ 点变轨进入椭圆轨道, 近地点 $B$ 到地心距离为 $r_{2}$ 。如图所示设卫星由 $A$ 到 $B$ (只经 $B$ 点一次) 运动的时间为 $t$, 地球自转周期为 $T_{0}$, 不计空气阻力,则 ( ) [图1] A: $T=\frac{3 T_{0}}{8}$ B: $T=\frac{3 T_{0}}{5}$ C: $t=\frac{\left(r_{1}+r_{2}\right) T}{4 r_{1}} \sqrt{\frac{r_{1}+r_{2}}{2 r_{1}}}$ D: $t=\frac{\left(r_{1}+r_{2}\right) T}{6 r_{1}} \sqrt{\frac{r_{1}+r_{2}}{2 r_{1}}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 已知某卫星在赤道上空轨道半径为 $r_{1}$ 的圆形轨道上绕地球运行的周期为 $T$, 卫星运动方向与地球自转方向相同, 赤道上某城市的人每三天恰好五次看到该卫星掠过其正上方。假设某时刻该卫星在 $A$ 点变轨进入椭圆轨道, 近地点 $B$ 到地心距离为 $r_{2}$ 。如图所示设卫星由 $A$ 到 $B$ (只经 $B$ 点一次) 运动的时间为 $t$, 地球自转周期为 $T_{0}$, 不计空气阻力,则 ( ) [图1] A: $T=\frac{3 T_{0}}{8}$ B: $T=\frac{3 T_{0}}{5}$ C: $t=\frac{\left(r_{1}+r_{2}\right) T}{4 r_{1}} \sqrt{\frac{r_{1}+r_{2}}{2 r_{1}}}$ D: $t=\frac{\left(r_{1}+r_{2}\right) T}{6 r_{1}} \sqrt{\frac{r_{1}+r_{2}}{2 r_{1}}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_475

宇宙中某一质量为 $M$ 、半径为 $R$ 的星球, 有三颗卫星 $\mathrm{A} 、 \mathrm{~B} 、 \mathrm{C}$ 在同一平面上沿逆时针方向做匀速圆周运动, 其位置关系如图所示。其中 $\mathrm{A}$ 到该星球表面的高度为 $h$, 已知引力常量为 $G$, 则下列说法正确的是 ( ) [图1] A: 三颗卫星的向心加速度大小关系为 $a_{\mathrm{A}}<a_{\mathrm{B}}=a_{\mathrm{C}}$ B: 三颗卫星的线速度大小关系为心 $v_{\mathrm{A}}>v_{\mathrm{B}}=v_{\mathrm{C}}$ C: 卫星 $\mathrm{C}$ 加速后可以追到卫星 $\mathrm{B}$ D: 卫星 $\mathrm{A}$ 的公转周期为 $2 \pi \sqrt{\frac{h^{3}}{G M}}$

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: 宇宙中某一质量为 $M$ 、半径为 $R$ 的星球, 有三颗卫星 $\mathrm{A} 、 \mathrm{~B} 、 \mathrm{C}$ 在同一平面上沿逆时针方向做匀速圆周运动, 其位置关系如图所示。其中 $\mathrm{A}$ 到该星球表面的高度为 $h$, 已知引力常量为 $G$, 则下列说法正确的是 ( ) [图1] A: 三颗卫星的向心加速度大小关系为 $a_{\mathrm{A}}<a_{\mathrm{B}}=a_{\mathrm{C}}$ B: 三颗卫星的线速度大小关系为心 $v_{\mathrm{A}}>v_{\mathrm{B}}=v_{\mathrm{C}}$ C: 卫星 $\mathrm{C}$ 加速后可以追到卫星 $\mathrm{B}$ D: 卫星 $\mathrm{A}$ 的公转周期为 $2 \pi \sqrt{\frac{h^{3}}{G M}}$ 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]

SC

Astronomy

ZH

multi-modal

Astronomy_751

What is the name of Jupiter's moon shown in the figure below? [figure1] A: Io B: Europa C: Callisto D: Ganymede

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 name of Jupiter's moon shown in the figure below? [figure1] A: Io B: Europa C: Callisto D: Ganymede 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

multi-modal

Astronomy_840

A comet passes near the Sun on a parabolic orbit. While it's passing near the Sun with orbital velocity $V$, the Sun's heat causes the comet to melt, and it shatters into many small fragments. The fragments move away uniformly in all directions (in the comet's reference frame) with velocity $v \ll V$. What fraction of the fragments will escape the solar system? Ignore any forces other than the Sun's gravity. A: $0 \%$ B: $50 \%$ C: $100 \%$ D: $\frac{v}{V}$ E: $1-\frac{v}{V}$

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 passes near the Sun on a parabolic orbit. While it's passing near the Sun with orbital velocity $V$, the Sun's heat causes the comet to melt, and it shatters into many small fragments. The fragments move away uniformly in all directions (in the comet's reference frame) with velocity $v \ll V$. What fraction of the fragments will escape the solar system? Ignore any forces other than the Sun's gravity. A: $0 \%$ B: $50 \%$ C: $100 \%$ D: $\frac{v}{V}$ E: $1-\frac{v}{V}$ 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_1072

A day on Earth can be defined in two ways: relative to the Sun (called solar or synodic time) or relative to the background stars (called sidereal time). The mean solar day is 24 hours (within a few milliseconds), whilst the mean sidereal day is shorter at 23 hours 56 minutes 4 seconds (to the nearest second). The solar day is longer as over the course of a sidereal day the Earth has moved slightly in its orbit around the Sun and so has to rotate slightly further for the Sun to be back in the same direction (see Figure 4). [figure1] Figure 4: A solar day is defined as the time between two consecutive passages of the Sun through the meridian, corresponding to local midday (which in the Northern hemisphere is in the South), whilst a sidereal day is the time for a distant star to do the same. The difference between the two is due to the Earth having moved slightly in its orbit around the Sun. Credit: Wikipedia. The length of a year on Earth is 365.25 solar days (to 2 d.p.), however some ancient civilizations used to believe that there were once exactly 360 solar days in a year, with various myths explaining where the extra days came from. In this question you will look at how to return the Earth to this time. [Note that this question is very sensitive to the precision of the fundamental constants used, so throughout please take $G=6.674 \times 10^{-11} \mathrm{~m}^{3} \mathrm{~kg}^{-1} \mathrm{~s}^{-2}, R_{\oplus}=6371 \mathrm{~km}, M_{\oplus}=5.972 \times 10^{24} \mathrm{~kg}, M_{\odot}=$ $1.989 \times 10^{30} \mathrm{~kg}$ and $1 \mathrm{au}=1.496 \times 10^{11} \mathrm{~m}$.] ## Helpful equations: The moment of inertia, $I$, of a sphere of mass $M$ and radius $R$ is $I=\frac{2}{5} M R^{2}$. 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. The speed, $v$, of an object in an elliptical orbit of semi-major axis $a$ around an object of mass $M$ when a distance $r$ away can be calculated as $$ v^{2}=G M\left(\frac{2}{r}-\frac{1}{a}\right) $$d. Imagine creating an incredibly powerful rocket, positioned on the Earth's equator, that when fired once can apply a huge force to the Earth in a very short time period, delivering a total impulse of $\Delta p$. Assuming the Earth's orbit is initially circular, calculate: i. The total impulse required to slow the Earth's rotation to give a year of 360 solar days, but with no change in the orbit.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: A day on Earth can be defined in two ways: relative to the Sun (called solar or synodic time) or relative to the background stars (called sidereal time). The mean solar day is 24 hours (within a few milliseconds), whilst the mean sidereal day is shorter at 23 hours 56 minutes 4 seconds (to the nearest second). The solar day is longer as over the course of a sidereal day the Earth has moved slightly in its orbit around the Sun and so has to rotate slightly further for the Sun to be back in the same direction (see Figure 4). [figure1] Figure 4: A solar day is defined as the time between two consecutive passages of the Sun through the meridian, corresponding to local midday (which in the Northern hemisphere is in the South), whilst a sidereal day is the time for a distant star to do the same. The difference between the two is due to the Earth having moved slightly in its orbit around the Sun. Credit: Wikipedia. The length of a year on Earth is 365.25 solar days (to 2 d.p.), however some ancient civilizations used to believe that there were once exactly 360 solar days in a year, with various myths explaining where the extra days came from. In this question you will look at how to return the Earth to this time. [Note that this question is very sensitive to the precision of the fundamental constants used, so throughout please take $G=6.674 \times 10^{-11} \mathrm{~m}^{3} \mathrm{~kg}^{-1} \mathrm{~s}^{-2}, R_{\oplus}=6371 \mathrm{~km}, M_{\oplus}=5.972 \times 10^{24} \mathrm{~kg}, M_{\odot}=$ $1.989 \times 10^{30} \mathrm{~kg}$ and $1 \mathrm{au}=1.496 \times 10^{11} \mathrm{~m}$.] ## Helpful equations: The moment of inertia, $I$, of a sphere of mass $M$ and radius $R$ is $I=\frac{2}{5} M R^{2}$. 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. The speed, $v$, of an object in an elliptical orbit of semi-major axis $a$ around an object of mass $M$ when a distance $r$ away can be calculated as $$ v^{2}=G M\left(\frac{2}{r}-\frac{1}{a}\right) $$ problem: d. Imagine creating an incredibly powerful rocket, positioned on the Earth's equator, that when fired once can apply a huge force to the Earth in a very short time period, delivering a total impulse of $\Delta p$. Assuming the Earth's orbit is initially circular, calculate: i. The total impulse required to slow the Earth's rotation to give a year of 360 solar days, but with no change in the orbit. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of \mathrm{~kg} \mathrm{~m} \mathrm{~s}^{-1}, 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_864

Suppose you are in Houston $\left(29^{\circ} 46^{\prime} N 95^{\circ} 23^{\prime} W\right)$ on the fall equinox and you just observed Deneb culminating (upper culmination). Knowing the data in the table of exercise 24, what is the hour angle of the Sun? A: 8h41min B: 20h41min C: $12 \mathrm{~h} 00 \mathrm{~min}$ D: $14 \mathrm{~h} 19 \mathrm{~min}$ E: $18 \mathrm{~h} 22 \mathrm{~min}$

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: Suppose you are in Houston $\left(29^{\circ} 46^{\prime} N 95^{\circ} 23^{\prime} W\right)$ on the fall equinox and you just observed Deneb culminating (upper culmination). Knowing the data in the table of exercise 24, what is the hour angle of the Sun? A: 8h41min B: 20h41min C: $12 \mathrm{~h} 00 \mathrm{~min}$ D: $14 \mathrm{~h} 19 \mathrm{~min}$ E: $18 \mathrm{~h} 22 \mathrm{~min}$ 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_380

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

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 如图所示, 质量分别为 $M$ 和 $m$ 的两个星球 $\mathrm{A}$ 和 $\mathrm{B}$ (均视为质点) 在它们之间的引力作用下都绕 $O$ 点做匀速圆周运动, 星球 $\mathrm{A}$ 和 $\mathrm{B}$ 之间的距离为 $L$ 。已知星球 $\mathrm{A}$ 和 $\mathrm{B}$ 和 $O$三点始终共线, $\mathrm{A}$ 和 $\mathrm{B}$ 分别在 $O$ 点的两侧。引力常量为 $G$ 。 求星球 $\mathrm{A}$ 的周期 $T$; [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_559

放置在水平平台上的物体, 其表观重力在数值上等于物体对平台的压力, 方向与压力的方向相同。微重力环境是指系统内物体的表观重力远小于其实际重力（万有引力）的环境。此环境下，物体的表观重力与其质量之比称为微重力加速度。 如图所示, 中国科学院力学研究所微重力实验室落塔是我国微重力实验的主要设施之一, 实验中落舱可采用单舱和双舱两种模式进行。已知地球表面的重力加速度为 $g$; 环绕地球做匀速圆周运动的人造卫星内部也存在微重力环境. 其产生原因简单来说是由于卫星实验舱不能被看作质点造成的, 只有在卫星的质心（质点系的质量中心）位置, 万有引力才恰好等于向心力. 已知某卫星绕地球做匀速圆周运动, 其质心到地心的距离为 $r$, 假设卫星实验舱中各点绕图中地球运动的角速度均与质心一致, 求 $g_{4}$ 与该卫星质心处的向心加速度 $a_{n}$ 的比值。 [图1] 落塔 落舱

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: 放置在水平平台上的物体, 其表观重力在数值上等于物体对平台的压力, 方向与压力的方向相同。微重力环境是指系统内物体的表观重力远小于其实际重力（万有引力）的环境。此环境下，物体的表观重力与其质量之比称为微重力加速度。 如图所示, 中国科学院力学研究所微重力实验室落塔是我国微重力实验的主要设施之一, 实验中落舱可采用单舱和双舱两种模式进行。已知地球表面的重力加速度为 $g$; 环绕地球做匀速圆周运动的人造卫星内部也存在微重力环境. 其产生原因简单来说是由于卫星实验舱不能被看作质点造成的, 只有在卫星的质心（质点系的质量中心）位置, 万有引力才恰好等于向心力. 已知某卫星绕地球做匀速圆周运动, 其质心到地心的距离为 $r$, 假设卫星实验舱中各点绕图中地球运动的角速度均与质心一致, 求 $g_{4}$ 与该卫星质心处的向心加速度 $a_{n}$ 的比值。 [图1] 落塔 落舱 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_361

宇宙中存在一些离其他恒星较远的三星系统, 通常可忽略其他星体对它们的引力作用, 现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, 两颗星围绕中央星做圆周运动, 如图甲所示; 另一种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, 如图乙所示, 设两种系统中三个星体的质量及各星间的距离如图甲、乙中所示, 已知引力常量为 $G$, 试分别求出两个系统做圆周运动的周期。 [图1] 甲 乙

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题包含多个待求解的量。 问题: 宇宙中存在一些离其他恒星较远的三星系统, 通常可忽略其他星体对它们的引力作用, 现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, 两颗星围绕中央星做圆周运动, 如图甲所示; 另一种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, 如图乙所示, 设两种系统中三个星体的质量及各星间的距离如图甲、乙中所示, 已知引力常量为 $G$, 试分别求出两个系统做圆周运动的周期。 [图1] 甲 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。 你的最终解答的量应该按以下顺序输出：[甲系统做圆周运动的周期, 乙系统做圆周运动的周期] 它们的答案类型依次是[表达式, 表达式] 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应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5

MPV

Astronomy

ZH

multi-modal

Astronomy_176

宇航员来到某星球表面做了如下实验: 将一小钢球由距星球表面高 $h(h$ 远小于星球半径）处由静止释放, 小钢球经过时间 $t$ 落到星球表面, 该星球为密度均匀的球体,引力常量为 $G$ 。 若该星球的半径为 $R$, 有一颗卫星在距该星球表面高度为 $H$ 处的圆轨道上绕该星球做匀速圆周运动, 求该卫星的线速度大小和周期。

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题包含多个待求解的量。 问题: 宇航员来到某星球表面做了如下实验: 将一小钢球由距星球表面高 $h(h$ 远小于星球半径）处由静止释放, 小钢球经过时间 $t$ 落到星球表面, 该星球为密度均匀的球体,引力常量为 $G$ 。 若该星球的半径为 $R$, 有一颗卫星在距该星球表面高度为 $H$ 处的圆轨道上绕该星球做匀速圆周运动, 求该卫星的线速度大小和周期。 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。 你的最终解答的量应该按以下顺序输出：[线速度大小, 周期] 它们的答案类型依次是[表达式, 表达式] 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应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5

MPV

Astronomy

ZH

text-only

Astronomy_496

假设宇宙是一团球形的密度均匀的物质, 其各物理量均具有球对称性（即只与球的半径有关)。宇宙球对称地向外膨胀, 半径为 $r$ 的位置具有速度 $v(r)$ 。不难发现, 宇宙膨胀的过程中, 其平均密度必然下降。若假设该宇宙球在膨胀过程中密度均匀（即球内各处密度相等), 则应该有 $v=\mathrm{H} r^{\alpha}$, 其中 $\mathrm{H}$ 是一个可变化但与 $r$ 无关的系数, 那么 $\alpha$ 的值应为 ( ) [提示: 若 $p(t)$ 是某一物理量, 则 $p^{a}$ 对时间的导数为 $a p^{a-1} p^{\prime}(t)$ ] A: 1 B: 2 C: 3 D: 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: 假设宇宙是一团球形的密度均匀的物质, 其各物理量均具有球对称性（即只与球的半径有关)。宇宙球对称地向外膨胀, 半径为 $r$ 的位置具有速度 $v(r)$ 。不难发现, 宇宙膨胀的过程中, 其平均密度必然下降。若假设该宇宙球在膨胀过程中密度均匀（即球内各处密度相等), 则应该有 $v=\mathrm{H} r^{\alpha}$, 其中 $\mathrm{H}$ 是一个可变化但与 $r$ 无关的系数, 那么 $\alpha$ 的值应为 ( ) [提示: 若 $p(t)$ 是某一物理量, 则 $p^{a}$ 对时间的导数为 $a p^{a-1} p^{\prime}(t)$ ] A: 1 B: 2 C: 3 D: 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]

SC

Astronomy

ZH

text-only

Astronomy_327

发射地球同步卫星时, 先将卫星发射至近地圆轨道 1, 然后经点火, 使其沿椭圆轨道 2 运行, 最后再次点火, 将卫星送入同步圆轨道 3. 轨道 $1 、 2$ 相切于 $Q$ 点, 轨道 $2 、 3$相切于 $P$ 点, 如图所示, 则当卫星分别在 $1 、 2 、 3$ 轨道上正常运行时, 以下说法不正确的是 ( ) [图1] A: 要将卫星由圆轨道 1 送入圆轨道 3 , 需要在圆轨道 1 的 $Q$ 点和椭圆轨道 2 的远地点 $P$ 分别点火加速一次 B: 由于卫星由圆轨道 1 送入圆轨道 3 点火加速两次, 则卫星在圆轨道 3 上正常运行速度大于卫星在圆轨道 1 上正常运行速度 C: 卫星在椭圆轨道 2 上的近地点 $Q$ 的速度一定大于 $7.9 \mathrm{~km} / \mathrm{s}$, 而在远地点 $P$ 的速度一定小于 $7.9 \mathrm{~km} / \mathrm{s}$ D: 卫星在椭圆轨道 2 上经过 $P$ 点时的加速度一定等于它在圆轨道 3 上经过 $P$ 点时的加速度

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: 发射地球同步卫星时, 先将卫星发射至近地圆轨道 1, 然后经点火, 使其沿椭圆轨道 2 运行, 最后再次点火, 将卫星送入同步圆轨道 3. 轨道 $1 、 2$ 相切于 $Q$ 点, 轨道 $2 、 3$相切于 $P$ 点, 如图所示, 则当卫星分别在 $1 、 2 、 3$ 轨道上正常运行时, 以下说法不正确的是 ( ) [图1] A: 要将卫星由圆轨道 1 送入圆轨道 3 , 需要在圆轨道 1 的 $Q$ 点和椭圆轨道 2 的远地点 $P$ 分别点火加速一次 B: 由于卫星由圆轨道 1 送入圆轨道 3 点火加速两次, 则卫星在圆轨道 3 上正常运行速度大于卫星在圆轨道 1 上正常运行速度 C: 卫星在椭圆轨道 2 上的近地点 $Q$ 的速度一定大于 $7.9 \mathrm{~km} / \mathrm{s}$, 而在远地点 $P$ 的速度一定小于 $7.9 \mathrm{~km} / \mathrm{s}$ D: 卫星在椭圆轨道 2 上经过 $P$ 点时的加速度一定等于它在圆轨道 3 上经过 $P$ 点时的加速度 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]

SC

Astronomy

ZH

multi-modal

Astronomy_1137

It is often said that the Sun rises in the East and sets in the West, however this is only true twice a year at the equinoxes. In the Northern hemisphere, the Sun will rise northwards of East on the June solstice, and southwards of East on the December solstice; this is directly tied in with the varying length of day too, since the Sun either has a greater or shorter distance to travel across the sky (see Figure 1). [figure1] Figure 1: Left: The path of the Sun across the sky during the equinoxes and solstices, as viewed by an observer in the Northern hemisphere at a latitude of $\sim 40^{\circ}$. Credit: Daniel V. Schroeder / Weber State University. Right: The same idea but viewed from Iceland at a latitude of $65^{\circ}$, where by being so close to the Artic circle the day length can get close to 24 hours in June and almost no daylight in December. Credit: Kristn Bjarnadttir / University of Iceland. During the equinox, the Sun travels along the projection of the Earth's equator. In this question, we will assume a circular orbit for the Earth, and all angles will be calculated in degrees. A simple model for the vertical angle between the Sun and the horizon (known at the altitude), $h$, as a function of the bearing on the horizon, $A$ (measured clockwise from North, also called the azimuth), the latitude of the observer, $\phi$ (positive in Northern hemisphere, negative in Southern hemisphere), and the vertical angle of the Sun relative to the celestial equator (known as the solar declination), $\delta$, is given as: $$ h=-\left(90^{\circ}-\phi\right) \cos (A)+\delta $$ The solar declination can be considered to vary sinusoidally over the year, going from a maximum of $\delta=+23.44^{\circ}$ at the June solstice (roughly $21^{\text {st }}$ June) to a minimum of $\delta=-23.44^{\circ}$ on the December solstice (roughly $21^{\text {st }}$ December). It can be shown using spherical trigonometry that the precise model connecting $\delta, h, \phi$ and $A$ is: $$ \sin (\delta)=\sin (h) \sin (\phi)+\cos (h) \cos (\phi) \cos (A) . $$ Using the precise model, the path of the Sun across the sky forms a shape that is not quite the cosine shape of the simple model, and is shown in Figure 2. [figure2] Figure 2: The altitude of the Sun as a function of bearing during the equinoxes and solstices, as viewed by an observer at a latitude of $+56^{\circ}$. Whilst it resembles the cosine shape of the simple model well at this latitude, there are small deviations. Credit: Wikipedia. By using further spherical trigonometry, we can derive a second helpful equation in the precise model: $$ \sin (h)=\sin (\phi) \sin (\delta)+\cos (\phi) \cos (\delta) \cos (H) $$ Here, $H$ is the solar hour angle, which measures the angle between the Sun and solar noon as measured along the projection of the Earth's equator on the sky. Conventionally, $H=0^{\circ}$ at solar noon, is negative before solar noon, and is positive afterwards. Since the sun's hour angle increases at an approximately constant rate due to the rotation of the Earth, we can convert this angle into a time using the conversion $360^{\circ}=24^{\mathrm{h}}$.c. Reconsider the Oxford observer at the June solstice, but this time use the two equations of the precise model. Ignore any atmospheric effects. ii. Calculate the duration of sunrise (in minutes and seconds), assuming a solar angular diameter of $0.525^{\circ}$.

You are participating in an international Astronomy competition and need to solve the following question. This question involves multiple quantities to be determined. Here is some context information for this question, which might assist you in solving it: It is often said that the Sun rises in the East and sets in the West, however this is only true twice a year at the equinoxes. In the Northern hemisphere, the Sun will rise northwards of East on the June solstice, and southwards of East on the December solstice; this is directly tied in with the varying length of day too, since the Sun either has a greater or shorter distance to travel across the sky (see Figure 1). [figure1] Figure 1: Left: The path of the Sun across the sky during the equinoxes and solstices, as viewed by an observer in the Northern hemisphere at a latitude of $\sim 40^{\circ}$. Credit: Daniel V. Schroeder / Weber State University. Right: The same idea but viewed from Iceland at a latitude of $65^{\circ}$, where by being so close to the Artic circle the day length can get close to 24 hours in June and almost no daylight in December. Credit: Kristn Bjarnadttir / University of Iceland. During the equinox, the Sun travels along the projection of the Earth's equator. In this question, we will assume a circular orbit for the Earth, and all angles will be calculated in degrees. A simple model for the vertical angle between the Sun and the horizon (known at the altitude), $h$, as a function of the bearing on the horizon, $A$ (measured clockwise from North, also called the azimuth), the latitude of the observer, $\phi$ (positive in Northern hemisphere, negative in Southern hemisphere), and the vertical angle of the Sun relative to the celestial equator (known as the solar declination), $\delta$, is given as: $$ h=-\left(90^{\circ}-\phi\right) \cos (A)+\delta $$ The solar declination can be considered to vary sinusoidally over the year, going from a maximum of $\delta=+23.44^{\circ}$ at the June solstice (roughly $21^{\text {st }}$ June) to a minimum of $\delta=-23.44^{\circ}$ on the December solstice (roughly $21^{\text {st }}$ December). It can be shown using spherical trigonometry that the precise model connecting $\delta, h, \phi$ and $A$ is: $$ \sin (\delta)=\sin (h) \sin (\phi)+\cos (h) \cos (\phi) \cos (A) . $$ Using the precise model, the path of the Sun across the sky forms a shape that is not quite the cosine shape of the simple model, and is shown in Figure 2. [figure2] Figure 2: The altitude of the Sun as a function of bearing during the equinoxes and solstices, as viewed by an observer at a latitude of $+56^{\circ}$. Whilst it resembles the cosine shape of the simple model well at this latitude, there are small deviations. Credit: Wikipedia. By using further spherical trigonometry, we can derive a second helpful equation in the precise model: $$ \sin (h)=\sin (\phi) \sin (\delta)+\cos (\phi) \cos (\delta) \cos (H) $$ Here, $H$ is the solar hour angle, which measures the angle between the Sun and solar noon as measured along the projection of the Earth's equator on the sky. Conventionally, $H=0^{\circ}$ at solar noon, is negative before solar noon, and is positive afterwards. Since the sun's hour angle increases at an approximately constant rate due to the rotation of the Earth, we can convert this angle into a time using the conversion $360^{\circ}=24^{\mathrm{h}}$. problem: c. Reconsider the Oxford observer at the June solstice, but this time use the two equations of the precise model. Ignore any atmospheric effects. ii. Calculate the duration of sunrise (in minutes and seconds), assuming a solar angular diameter of $0.525^{\circ}$. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Your final quantities should be output in the following order: [H_{-}, H_{+}]. Their units are, in order, [\circ, \circ], but units shouldn't be included in your concluded answer. Their answer types are, in order, [numerical value, numerical value]. Please end your response with: "The final answers are \boxed{ANSWER}", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5

MPV

Astronomy

EN

multi-modal

Astronomy_836

After that slight headache, Austin is back at MIT in Boston! For his astronomy research, he is observing the LARES satellite which is a ball of diameter $36.4 \mathrm{~cm}$ made out of THA-18N (a tungsten alloy). It orbits at a distance $1450 \mathrm{~km}$ from the surface of the Earth and at an inclination of $69.49^{\circ}$ relative to the equatorial plane. What is the highest altitude Austin can point his telescope if he wants to observe LARES at its highest latitude? A: $9.4^{\circ}$ B: $14.4^{\circ}$ C: $18.4^{\circ}$ D: $23.4^{\circ}$ E: $33.4^{\circ}$

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: After that slight headache, Austin is back at MIT in Boston! For his astronomy research, he is observing the LARES satellite which is a ball of diameter $36.4 \mathrm{~cm}$ made out of THA-18N (a tungsten alloy). It orbits at a distance $1450 \mathrm{~km}$ from the surface of the Earth and at an inclination of $69.49^{\circ}$ relative to the equatorial plane. What is the highest altitude Austin can point his telescope if he wants to observe LARES at its highest latitude? A: $9.4^{\circ}$ B: $14.4^{\circ}$ C: $18.4^{\circ}$ D: $23.4^{\circ}$ E: $33.4^{\circ}$ 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_34

2021 年 5 月，基于俗称“中国天眼”的 500 米口径球面射电望远镜（FAST）的观测,国家天文台李䓎、朱炜玮研究团组姚菊枚博士等首次研究发现脉冲星三维速度与自转轴共线的证据。之前的 2020 年 3 月, 我国天文学家通过 FAST, 在武仙座球状星团 （M13）中发现一个脉冲双星系统。如图所示，假设在太空中有恒星 A、B 双星系统绕点 $O$ 做顺时针匀速圆周运动, 运动周期为 $T_{1}$, 它们的轨道半径分别为 $R_{A} 、 R_{B}, R_{A}<$ $R_{B}, \mathrm{C}$ 为 $\mathrm{B}$ 的卫星, 绕 $\mathrm{B}$ 做逆时针匀速圆周运动, 周期为 $T_{2}$, 忽略 $\mathrm{A}$ 与 $\mathrm{C}$ 之间的引力,万引力常量为 $G$, 则以下说法正确的是 ( ) [图1] A: 若知道 $\mathrm{C}$ 的轨道半径, 则可求出 $\mathrm{C}$ 的质量 B: 恒星 $\mathrm{A} 、 \mathrm{~B}$ 的质量和为 $\frac{4 \pi^{2}\left(R_{\mathrm{A}}+R_{\mathrm{B}}\right)^{3}}{G T_{1}^{2}}$ C: 若 $\mathrm{A}$ 也有一颗运动周期为 $\mathrm{T}_{2}$ 的卫星, 则其轨道半径大于 $\mathrm{C}$ 的轨道半径 D: 设 $\mathrm{A} 、 \mathrm{~B} 、 \mathrm{C}$ 三星由图示位置到再次共线的时间为 $t$, 则 $t=\frac{T_{1} T_{2}}{\left.2 T_{1}+T_{2}\right)}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 2021 年 5 月，基于俗称“中国天眼”的 500 米口径球面射电望远镜（FAST）的观测,国家天文台李䓎、朱炜玮研究团组姚菊枚博士等首次研究发现脉冲星三维速度与自转轴共线的证据。之前的 2020 年 3 月, 我国天文学家通过 FAST, 在武仙座球状星团 （M13）中发现一个脉冲双星系统。如图所示，假设在太空中有恒星 A、B 双星系统绕点 $O$ 做顺时针匀速圆周运动, 运动周期为 $T_{1}$, 它们的轨道半径分别为 $R_{A} 、 R_{B}, R_{A}<$ $R_{B}, \mathrm{C}$ 为 $\mathrm{B}$ 的卫星, 绕 $\mathrm{B}$ 做逆时针匀速圆周运动, 周期为 $T_{2}$, 忽略 $\mathrm{A}$ 与 $\mathrm{C}$ 之间的引力,万引力常量为 $G$, 则以下说法正确的是 ( ) [图1] A: 若知道 $\mathrm{C}$ 的轨道半径, 则可求出 $\mathrm{C}$ 的质量 B: 恒星 $\mathrm{A} 、 \mathrm{~B}$ 的质量和为 $\frac{4 \pi^{2}\left(R_{\mathrm{A}}+R_{\mathrm{B}}\right)^{3}}{G T_{1}^{2}}$ C: 若 $\mathrm{A}$ 也有一颗运动周期为 $\mathrm{T}_{2}$ 的卫星, 则其轨道半径大于 $\mathrm{C}$ 的轨道半径 D: 设 $\mathrm{A} 、 \mathrm{~B} 、 \mathrm{C}$ 三星由图示位置到再次共线的时间为 $t$, 则 $t=\frac{T_{1} T_{2}}{\left.2 T_{1}+T_{2}\right)}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_636

如图所示，宇宙飞船绕地球做圆周运动时，由于地球遮挡阳光，会经历“日全食”过 程, 太阳光可看作平行光, 宇航员在 $A$ 点测出地球的张角为 $\alpha$ 。已知地球半径为 $R$, 地 球质量为 $\mathrm{M}$, 引力常量为 $G$, 不考虑地球公转的影响。求: 飞船运行的高度 $h$; [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，宇宙飞船绕地球做圆周运动时，由于地球遮挡阳光，会经历“日全食”过 程, 太阳光可看作平行光, 宇航员在 $A$ 点测出地球的张角为 $\alpha$ 。已知地球半径为 $R$, 地 球质量为 $\mathrm{M}$, 引力常量为 $G$, 不考虑地球公转的影响。求: 飞船运行的高度 $h$; [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_945

The One-Mile Telescope is a radio telescope just outside Cambridge that has three dishes that can be spread out up to one mile $(=1.6 \mathrm{~km})$ apart. Two of the dishes are fixed, whilst one can move along an $800-\mathrm{m}$ set of former railway tracks. In order for the tracks to be perfectly flat, how much did they need to raise one end to compensate for the curvature of the Earth? A: $5 \mathrm{~cm}$ B: $10 \mathrm{~cm}$ C: $15 \mathrm{~cm}$ D: $20 \mathrm{~cm}$

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 One-Mile Telescope is a radio telescope just outside Cambridge that has three dishes that can be spread out up to one mile $(=1.6 \mathrm{~km})$ apart. Two of the dishes are fixed, whilst one can move along an $800-\mathrm{m}$ set of former railway tracks. In order for the tracks to be perfectly flat, how much did they need to raise one end to compensate for the curvature of the Earth? A: $5 \mathrm{~cm}$ B: $10 \mathrm{~cm}$ C: $15 \mathrm{~cm}$ D: $20 \mathrm{~cm}$ 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_470

在 X 星球表面, 宇航员做了一个实验：如图甲所示, 轻杆一端固定在 $O$ 点, 另一端固定一小球，现让小球在坚直平面内做半径为 $R$ 的圆周运动。小球运动到最高点时,受到的弹力为 $F$, 速度大小为 $v$, 其 $F-v^{2}$ 图像如乙图所示。已知 $\mathrm{X}$ 星球的半径为 $R_{0}$, 万有引力常量为 $G$, 不考虑星球自转。则下列说法正确的是 ( ) [图1] 甲 [图2] 乙 A: X 星球的第一宇宙速度 $v_{1}=\sqrt{b}$ B: X 星球的密度 $\rho=\frac{3 b}{4 \pi G R_{0}}$ C: $\mathrm{X}$ 星球的质量 $M=\frac{b R_{0}^{2}}{G R}$ D: 环绕 $\mathrm{X}$ 星球的轨道离星球表面高度为 $R_{0}$ 的卫星周期 $T=2 \pi \sqrt{\frac{8 R R_{0}}{b}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 在 X 星球表面, 宇航员做了一个实验：如图甲所示, 轻杆一端固定在 $O$ 点, 另一端固定一小球，现让小球在坚直平面内做半径为 $R$ 的圆周运动。小球运动到最高点时,受到的弹力为 $F$, 速度大小为 $v$, 其 $F-v^{2}$ 图像如乙图所示。已知 $\mathrm{X}$ 星球的半径为 $R_{0}$, 万有引力常量为 $G$, 不考虑星球自转。则下列说法正确的是 ( ) [图1] 甲 [图2] 乙 A: X 星球的第一宇宙速度 $v_{1}=\sqrt{b}$ B: X 星球的密度 $\rho=\frac{3 b}{4 \pi G R_{0}}$ C: $\mathrm{X}$ 星球的质量 $M=\frac{b R_{0}^{2}}{G R}$ D: 环绕 $\mathrm{X}$ 星球的轨道离星球表面高度为 $R_{0}$ 的卫星周期 $T=2 \pi \sqrt{\frac{8 R R_{0}}{b}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_714

2018 年 10 月 20 日, 酒泉卫星发射中心迎来 60 岁生日. 作为我国航天事业的发祥地, 中心拥有我国最早的航天发射场和目前唯一的载人航天发射场. 2013 年 6 月, 我国成功实现目标飞行器“神舟十号”与轨道空间站“天宫一号” 的对接. 如图所示, 已知“神舟十号”从捕获“天宫一号”到两个飞行器实现刚性对接用时为 $t$, 这段时间内组合体绕地球转过的角度为 $\theta$, 地球半径为 $R$, 组合体离地面的高度为 $H$, 万有引力常量为 $G$, 据以上信息可求地球的质量为 [图1] “天宫一号”与 “神舟十号”成功实现自动交会对接 A: $\frac{(R+H)^{3} \theta^{2}}{G t^{2}}$ B: $\frac{\pi^{2}(R+H)^{3} \theta^{2}}{G t^{2}}$ C: $\frac{(R+H)^{3} \theta^{2}}{4 \pi G t^{2}}$ D: $\frac{4 \pi^{4}(R+H)^{3} \theta^{2}}{G t^{2}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 2018 年 10 月 20 日, 酒泉卫星发射中心迎来 60 岁生日. 作为我国航天事业的发祥地, 中心拥有我国最早的航天发射场和目前唯一的载人航天发射场. 2013 年 6 月, 我国成功实现目标飞行器“神舟十号”与轨道空间站“天宫一号” 的对接. 如图所示, 已知“神舟十号”从捕获“天宫一号”到两个飞行器实现刚性对接用时为 $t$, 这段时间内组合体绕地球转过的角度为 $\theta$, 地球半径为 $R$, 组合体离地面的高度为 $H$, 万有引力常量为 $G$, 据以上信息可求地球的质量为 [图1] “天宫一号”与 “神舟十号”成功实现自动交会对接 A: $\frac{(R+H)^{3} \theta^{2}}{G t^{2}}$ B: $\frac{\pi^{2}(R+H)^{3} \theta^{2}}{G t^{2}}$ C: $\frac{(R+H)^{3} \theta^{2}}{4 \pi G t^{2}}$ D: $\frac{4 \pi^{4}(R+H)^{3} \theta^{2}}{G t^{2}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

multi-modal

Astronomy_141

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。 物体间由于存在万有引力而具有的势能称为引力势能。若取两物体相距无穷远时引力势能为 0 , 质点 $m_{1}$ 和 $m_{2}$ 的距离为 $r$ 时, 其引力势能为 $E_{\mathrm{p}}=-\frac{G m_{1} m_{2}}{r}$ (式中 $G$ 为万有引力常量)。假设空间站为避免与其它飞行物相撞, 将从原轨道转移到距地球表面高为 $1.2 h$ 的新圆周轨道上, 则该转移至少需要提供多少额外的能量;

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。 物体间由于存在万有引力而具有的势能称为引力势能。若取两物体相距无穷远时引力势能为 0 , 质点 $m_{1}$ 和 $m_{2}$ 的距离为 $r$ 时, 其引力势能为 $E_{\mathrm{p}}=-\frac{G m_{1} m_{2}}{r}$ (式中 $G$ 为万有引力常量)。假设空间站为避免与其它飞行物相撞, 将从原轨道转移到距地球表面高为 $1.2 h$ 的新圆周轨道上, 则该转移至少需要提供多少额外的能量; 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_281

假设有一载人宇宙飞船在距地面高度为 $4200 \mathrm{~km}$ 的赤道上空绕地球做匀速圆周运动, 地球半径约为 $6400 \mathrm{~km}$, 地球同步卫星距地面高为 $36000 \mathrm{~km}$ 。宇宙飞船和一地球同步卫星绕地球同向运动, 每当两者相距最近时, 宇宙飞船就向同步卫星发射信号, 然后再由同步卫星将信号发送到地面接收站。某时刻两者相距最远, 从此刻开始, 在一昼夜的时间内，接收站共接收到信号的次数为 A: 7 次 B: 6 次 C: 5 次 D: 4 次

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 假设有一载人宇宙飞船在距地面高度为 $4200 \mathrm{~km}$ 的赤道上空绕地球做匀速圆周运动, 地球半径约为 $6400 \mathrm{~km}$, 地球同步卫星距地面高为 $36000 \mathrm{~km}$ 。宇宙飞船和一地球同步卫星绕地球同向运动, 每当两者相距最近时, 宇宙飞船就向同步卫星发射信号, 然后再由同步卫星将信号发送到地面接收站。某时刻两者相距最远, 从此刻开始, 在一昼夜的时间内，接收站共接收到信号的次数为 A: 7 次 B: 6 次 C: 5 次 D: 4 次 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_1222

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.c. Calculate $\beta_{\text {app }}$ for both jets, and use your formula from part $b$. to calculate the minimum value of $\beta$ to explain the apparent superluminal motion.

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: c. Calculate $\beta_{\text {app }}$ for both jets, and use your formula from part $b$. to calculate the minimum value of $\beta$ to explain the apparent superluminal motion. 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_722

两颗人造卫星绕地球逆时针运动, 卫星 1、卫星 2 分别沿圆轨道、椭圆轨道运动,圆的半径与粗圆的半长轴相等, 两轨道相交于 $A 、 B$ 两点, 某时刻两卫星与地球在同一直线上，如图所示，下列说法中正确的是（） [图1] A: 两卫星在图示位置的速度 $v_{2}=v_{1}$ B: 两卫星在 $A$ 处的加速度大小不相等 C: 两颗卫星在 $A$ 或 $B$ 点处可能相遇 D: 两卫星永远不可能相遇

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: 两颗人造卫星绕地球逆时针运动, 卫星 1、卫星 2 分别沿圆轨道、椭圆轨道运动,圆的半径与粗圆的半长轴相等, 两轨道相交于 $A 、 B$ 两点, 某时刻两卫星与地球在同一直线上，如图所示，下列说法中正确的是（） [图1] A: 两卫星在图示位置的速度 $v_{2}=v_{1}$ B: 两卫星在 $A$ 处的加速度大小不相等 C: 两颗卫星在 $A$ 或 $B$ 点处可能相遇 D: 两卫星永远不可能相遇 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_31

嫦娥四号”在月球背面软着陆和巡视探测, 创造了人类探月的历史. 为了实现“嫦娥四号”与地面间的太空通讯，我国于 2018 年 5 月发射了中继卫星“鹊桥”，它是运行于 地月拉格朗日 $\mathrm{L}_{2}$ 点的通信卫星, $\mathrm{L}_{2}$ 点位于地球和月球连线的延长线上. 若某飞行器位于 $\mathrm{L}_{2}$ 点, 可以在几乎不消耗燃料的情况下与月球同步绕地球做匀速圆周运动, 如图所示. 已知地球质量是月球质量的 $\mathrm{k}$ 倍, 飞行器质量远小于月球质量, 地球与月球中心距离是 $\mathrm{L}_{2}$ 点与月球中心距离的 $\mathrm{n}$ 倍. 下列说法正确的是 [图1] A: 飞行器的加速度大于月球的加速度 B: 飞行器的运行周期大于月球的运行周期 C: 飞行器所需的向心力由地球对其引力提供 D: $\mathrm{k}$ 与 $\mathrm{n}$ 满足 $\mathrm{k}=\frac{n^{3}(n+1)^{2}}{3 n^{2}+3 n+1}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 嫦娥四号”在月球背面软着陆和巡视探测, 创造了人类探月的历史. 为了实现“嫦娥四号”与地面间的太空通讯，我国于 2018 年 5 月发射了中继卫星“鹊桥”，它是运行于 地月拉格朗日 $\mathrm{L}_{2}$ 点的通信卫星, $\mathrm{L}_{2}$ 点位于地球和月球连线的延长线上. 若某飞行器位于 $\mathrm{L}_{2}$ 点, 可以在几乎不消耗燃料的情况下与月球同步绕地球做匀速圆周运动, 如图所示. 已知地球质量是月球质量的 $\mathrm{k}$ 倍, 飞行器质量远小于月球质量, 地球与月球中心距离是 $\mathrm{L}_{2}$ 点与月球中心距离的 $\mathrm{n}$ 倍. 下列说法正确的是 [图1] A: 飞行器的加速度大于月球的加速度 B: 飞行器的运行周期大于月球的运行周期 C: 飞行器所需的向心力由地球对其引力提供 D: $\mathrm{k}$ 与 $\mathrm{n}$ 满足 $\mathrm{k}=\frac{n^{3}(n+1)^{2}}{3 n^{2}+3 n+1}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_397

如图所示, 甲、乙分别为常见的三星系统模型和四星系统模型。甲图中三颗质量均为 $m$ 的行星都绕边长为 $L_{1}$ 的等边三角形的中心做匀速圆周运动, 周期为 $T_{1}$; 乙图中三 颗质量均为 $m$ 的行星都绕静止于边长为 $L_{2}$ 的等边三角形中心的中央星做匀速圆周运动,周期为 $T_{2}$, 不考虑其它星系的影响。已知四星系统内中央星的质量 $M=\sqrt{3} m$, $L_{2}=2 L_{1}$, 则两个系统的周期之比为 $(\quad)$ [图1] 甲 [图2] 乙 A: $T_{1}: T_{2}=1: 1$ B: $T_{1}: T_{2}=1: \sqrt{2}$ C: $T_{1}: T_{2}=1: \sqrt{3}$ D: $T_{1}: T_{2}=1: 2$

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, 甲、乙分别为常见的三星系统模型和四星系统模型。甲图中三颗质量均为 $m$ 的行星都绕边长为 $L_{1}$ 的等边三角形的中心做匀速圆周运动, 周期为 $T_{1}$; 乙图中三 颗质量均为 $m$ 的行星都绕静止于边长为 $L_{2}$ 的等边三角形中心的中央星做匀速圆周运动,周期为 $T_{2}$, 不考虑其它星系的影响。已知四星系统内中央星的质量 $M=\sqrt{3} m$, $L_{2}=2 L_{1}$, 则两个系统的周期之比为 $(\quad)$ [图1] 甲 [图2] 乙 A: $T_{1}: T_{2}=1: 1$ B: $T_{1}: T_{2}=1: \sqrt{2}$ C: $T_{1}: T_{2}=1: \sqrt{3}$ D: $T_{1}: T_{2}=1: 2$ 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

ZH

multi-modal

Astronomy_981

Which well-known star dropped in brightness by $40 \%$ between October 2019 and April 2020, leading to speculation it may be about to go supernova? A: Aldebaran B: Antares C: Arcturus D: Betelgeuse

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 well-known star dropped in brightness by $40 \%$ between October 2019 and April 2020, leading to speculation it may be about to go supernova? A: Aldebaran B: Antares C: Arcturus D: Betelgeuse 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_630

一卫星绕地球做椭圆轨道运动, 近地点距地表 $h_{1}=3600 \mathrm{~km}$, 远地点距地表 $h_{2}=23600 \mathrm{~km}$ 。假设在近地点卫星加速, 使得椭圆轨道的远地点距离地球表面 $h_{3}=33600 \mathrm{~km}$ 。已知地球半径 $r=6400 \mathrm{~km}$, 则卫星变轨时的速度增量应约为（）（设地球表面重力加速度 $g=10 \mathrm{~m} / \mathrm{s}^{2}$ )。 [图1] A: $1000 \mathrm{~m} / \mathrm{s}$ B: $500 \mathrm{~m} / \mathrm{s}$ C: $250 \mathrm{~m} / \mathrm{s}$ D: $100 \mathrm{~m} / \mathrm{s}$

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: 一卫星绕地球做椭圆轨道运动, 近地点距地表 $h_{1}=3600 \mathrm{~km}$, 远地点距地表 $h_{2}=23600 \mathrm{~km}$ 。假设在近地点卫星加速, 使得椭圆轨道的远地点距离地球表面 $h_{3}=33600 \mathrm{~km}$ 。已知地球半径 $r=6400 \mathrm{~km}$, 则卫星变轨时的速度增量应约为（）（设地球表面重力加速度 $g=10 \mathrm{~m} / \mathrm{s}^{2}$ )。 [图1] A: $1000 \mathrm{~m} / \mathrm{s}$ B: $500 \mathrm{~m} / \mathrm{s}$ C: $250 \mathrm{~m} / \mathrm{s}$ D: $100 \mathrm{~m} / \mathrm{s}$ 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]

SC

Astronomy

ZH

multi-modal

Astronomy_1132

Recent years have seen an explosion in the discovery of new exoplanets. About $85 \%$ of transiting exoplanets discovered by the NASA Kepler telescope have radii less than Neptune ( $\sim 4 R_{\oplus}$ ), meaning we are improving our understanding of what the transition between rocky Earth-size planets and gaseous Neptune-size planets looks like. Given how common these "super-Earths" and "gas dwarfs" seem to be, it was odd that we didn't have any in our own Solar System. However, Batygin \& Brown (2016) suggested that a hypothetical ninth planet (called 'Planet Nine') could explain some of the unusual properties of the orbits of objects in the Kuiper Belt. This planet is inferred to have a mass of $10 M_{\oplus}$, and so would be an example of a super-Earth. [figure1] Figure 5: A plot of planet density versus radius for 33 extrasolar planets (circles) and the planets in our solar system (diamonds). Credit: Marcy et al. (2014). Analysing exoplanets discovered by Kepler, Marcy et al. (2014) used a piecewise function to describe their planetary density data such that: $$ \begin{aligned} \text { For } R_{\mathrm{P}} \leq 1.5 R_{\oplus} & \rho & =2.32+3.18 \frac{R_{\mathrm{P}}}{R_{\oplus}}\left[\mathrm{g} \mathrm{cm}^{-3}\right] \\ \text { For } 1.5 R_{\oplus}<R_{\mathrm{P}} \leq 4.2 R_{\oplus} & \frac{M_{\mathrm{P}}}{M_{\oplus}} & =2.69\left(\frac{R_{\mathrm{P}}}{R_{\oplus}}\right)^{0.93} \end{aligned} $$ where $R_{\mathrm{P}}$ is the radius of the planet, $M_{\mathrm{P}}$ is the mass of the planet, and the model's transition between rocky super-Earth and non-rocky gas dwarf occurs at $R_{\mathrm{P}}=1.5 R_{\oplus}$. The minimum speed necessary to fully escape a planet's gravity (rather than be put into an elliptical orbit) is called the escape velocity and is calculated as $$ v_{\mathrm{esc}}=\sqrt{\frac{2 G M_{\mathrm{P}}}{R_{\mathrm{P}}}} $$ where $G$ is the universal gravitational constant. In contrast, the maximum speed a rocket can provide is determined by the ejection velocity, $v_{\mathrm{e}}$, of the gas used, as determined by the chemical energy stored in the bonds of the fuel used, and the fraction of the rocket that is fuel. Since the rocket gets lighter as it burns its fuel the final speed can be bigger than $v_{\mathrm{e}}$. The rocket equation is $$ v_{\max }=v_{\mathrm{e}} \ln \frac{m_{0}}{m_{1}} $$ where $m_{0}$ is the mass at launch and $m_{1}$ is the mass once all the fuel has been burnt. The most energetic chemical reaction we can use in a rocket is hydrogen-oxygen, which gives $v_{\mathrm{e}}=4.46 \mathrm{~km} \mathrm{~s}^{-1}$, and engineering limits us to a rocket design with a maximum of $96 \%$ of launch mass being fuel (as was used with the solid rockets that launched the space shuttle).a. Based on the Marcy et al. (2014) model, Planet Nine is most likely to be a gas dwarf with a thick gaseous envelope. Calculate $R_{P}$ (in units of $R_{E}$ ) for Planet Nine using this model.

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: Recent years have seen an explosion in the discovery of new exoplanets. About $85 \%$ of transiting exoplanets discovered by the NASA Kepler telescope have radii less than Neptune ( $\sim 4 R_{\oplus}$ ), meaning we are improving our understanding of what the transition between rocky Earth-size planets and gaseous Neptune-size planets looks like. Given how common these "super-Earths" and "gas dwarfs" seem to be, it was odd that we didn't have any in our own Solar System. However, Batygin \& Brown (2016) suggested that a hypothetical ninth planet (called 'Planet Nine') could explain some of the unusual properties of the orbits of objects in the Kuiper Belt. This planet is inferred to have a mass of $10 M_{\oplus}$, and so would be an example of a super-Earth. [figure1] Figure 5: A plot of planet density versus radius for 33 extrasolar planets (circles) and the planets in our solar system (diamonds). Credit: Marcy et al. (2014). Analysing exoplanets discovered by Kepler, Marcy et al. (2014) used a piecewise function to describe their planetary density data such that: $$ \begin{aligned} \text { For } R_{\mathrm{P}} \leq 1.5 R_{\oplus} & \rho & =2.32+3.18 \frac{R_{\mathrm{P}}}{R_{\oplus}}\left[\mathrm{g} \mathrm{cm}^{-3}\right] \\ \text { For } 1.5 R_{\oplus}<R_{\mathrm{P}} \leq 4.2 R_{\oplus} & \frac{M_{\mathrm{P}}}{M_{\oplus}} & =2.69\left(\frac{R_{\mathrm{P}}}{R_{\oplus}}\right)^{0.93} \end{aligned} $$ where $R_{\mathrm{P}}$ is the radius of the planet, $M_{\mathrm{P}}$ is the mass of the planet, and the model's transition between rocky super-Earth and non-rocky gas dwarf occurs at $R_{\mathrm{P}}=1.5 R_{\oplus}$. The minimum speed necessary to fully escape a planet's gravity (rather than be put into an elliptical orbit) is called the escape velocity and is calculated as $$ v_{\mathrm{esc}}=\sqrt{\frac{2 G M_{\mathrm{P}}}{R_{\mathrm{P}}}} $$ where $G$ is the universal gravitational constant. In contrast, the maximum speed a rocket can provide is determined by the ejection velocity, $v_{\mathrm{e}}$, of the gas used, as determined by the chemical energy stored in the bonds of the fuel used, and the fraction of the rocket that is fuel. Since the rocket gets lighter as it burns its fuel the final speed can be bigger than $v_{\mathrm{e}}$. The rocket equation is $$ v_{\max }=v_{\mathrm{e}} \ln \frac{m_{0}}{m_{1}} $$ where $m_{0}$ is the mass at launch and $m_{1}$ is the mass once all the fuel has been burnt. The most energetic chemical reaction we can use in a rocket is hydrogen-oxygen, which gives $v_{\mathrm{e}}=4.46 \mathrm{~km} \mathrm{~s}^{-1}$, and engineering limits us to a rocket design with a maximum of $96 \%$ of launch mass being fuel (as was used with the solid rockets that launched the space shuttle). problem: a. Based on the Marcy et al. (2014) model, Planet Nine is most likely to be a gas dwarf with a thick gaseous envelope. Calculate $R_{P}$ (in units of $R_{E}$ ) for Planet Nine using this model. 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_132

2016 年 10 月 19 日凌晨 3 点, 航天员景海鹏和陈冬驾驶的神舟十一号飞船与天宫二号空间实验室在离地面 $393 \mathrm{~km}$ 的近圆形轨道上成功实现了太空之吻。若对接轨道所处的空间存在极其稀薄的大气，则下面说法不正确的是（ ） A: 实现对接后, 组合体运行速度大于第一宇宙速度 B: 航天员景海鹏和陈冬能在天宫二号中自由飞翔, 说明他们不受地球引力作用 C: 如不加干预, 在运行一段时间后, 组合体的动能可能会增加 D: 如不加干预, 组合体的轨道高度将缓慢升高

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: 2016 年 10 月 19 日凌晨 3 点, 航天员景海鹏和陈冬驾驶的神舟十一号飞船与天宫二号空间实验室在离地面 $393 \mathrm{~km}$ 的近圆形轨道上成功实现了太空之吻。若对接轨道所处的空间存在极其稀薄的大气，则下面说法不正确的是（ ） A: 实现对接后, 组合体运行速度大于第一宇宙速度 B: 航天员景海鹏和陈冬能在天宫二号中自由飞翔, 说明他们不受地球引力作用 C: 如不加干预, 在运行一段时间后, 组合体的动能可能会增加 D: 如不加干预, 组合体的轨道高度将缓慢升高 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_1047

How far away must your friend be standing from you such that the attractive force exerted on you is similar to the maximum gravitational force exerted on you by Mars? Assume that your friend's mass is $65 \mathrm{~kg}$. The mass of Mars is $6.4 \times 10^{23} \mathrm{~kg}$ and the minimum distance from Earth to Mars is 0.52 AU. A: $2.3 \mathrm{~m}$ B: $0.8 \mathrm{~mm}$ C: $0.8 \mathrm{~m}$ D: $2.3 \mathrm{~mm}$

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: How far away must your friend be standing from you such that the attractive force exerted on you is similar to the maximum gravitational force exerted on you by Mars? Assume that your friend's mass is $65 \mathrm{~kg}$. The mass of Mars is $6.4 \times 10^{23} \mathrm{~kg}$ and the minimum distance from Earth to Mars is 0.52 AU. A: $2.3 \mathrm{~m}$ B: $0.8 \mathrm{~mm}$ C: $0.8 \mathrm{~m}$ D: $2.3 \mathrm{~mm}$ You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

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Astronomy

EN

text-only

Astronomy_603

2017 年, 人类第一次直接探测到来自双中子星合并的引力波 (引力波的周期与其 相互环绕的周期一致). 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: 质量之和 B: 速率之比 C: 速率之和 D: 所辐射出的引力波频率

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 (引力波的周期与其 相互环绕的周期一致). 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: 质量之和 B: 速率之比 C: 速率之和 D: 所辐射出的引力波频率 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_1076

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.d. For the planet WTS-2b, assuming $Q^{\prime}{ }_{\star}=10^{6}$ : ii. Show that $T_{\text {shift }} \propto \frac{T^{2}}{\tau}$, finding the constant of proportionality, and hence verify that $T_{\text {shift }}$ is measurable (i.e. $>5 \mathrm{~s}$ ) if $\mathrm{T}=10$ years. Hint: Using the chain rule, $\mathrm{dn} / \mathrm{dt}=\mathrm{dn} / \mathrm{da}$

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: 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: d. For the planet WTS-2b, assuming $Q^{\prime}{ }_{\star}=10^{6}$ : ii. Show that $T_{\text {shift }} \propto \frac{T^{2}}{\tau}$, finding the constant of proportionality, and hence verify that $T_{\text {shift }}$ is measurable (i.e. $>5 \mathrm{~s}$ ) if $\mathrm{T}=10$ years. Hint: Using the chain rule, $\mathrm{dn} / \mathrm{dt}=\mathrm{dn} / \mathrm{da}$ 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_577

如图所示, $a$ 为放在赤道上相对地球静止的物体, 随地球自转做匀速圆周运动, $b$为沿地面表面附近做匀速圆周运动的人造卫星 (轨道半径等于地球半径), $c$ 为地球的同步卫星，以下关于 $a 、 b 、 c$ 的说法中正确的是（） [图1] A: $a 、 b 、 c$ 做匀速圆周运动的向心加速度大小关系为 $a_{b}>a_{c}>a_{a}$ B: $a 、 b 、 c$ 做匀速圆周运动的角速度大小关系为 $\omega_{a}=\omega_{c}>\omega_{b}$ C: $a 、 b 、 c$ 做匀速圆周运动的线速度大小关系为 $v_{a}=v_{b}>v_{c}$ D: $a 、 b 、 c$ 做匀速圆周运动的周期关系为 $T_{a}=T_{c}>T_{b}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 如图所示, $a$ 为放在赤道上相对地球静止的物体, 随地球自转做匀速圆周运动, $b$为沿地面表面附近做匀速圆周运动的人造卫星 (轨道半径等于地球半径), $c$ 为地球的同步卫星，以下关于 $a 、 b 、 c$ 的说法中正确的是（） [图1] A: $a 、 b 、 c$ 做匀速圆周运动的向心加速度大小关系为 $a_{b}>a_{c}>a_{a}$ B: $a 、 b 、 c$ 做匀速圆周运动的角速度大小关系为 $\omega_{a}=\omega_{c}>\omega_{b}$ C: $a 、 b 、 c$ 做匀速圆周运动的线速度大小关系为 $v_{a}=v_{b}>v_{c}$ D: $a 、 b 、 c$ 做匀速圆周运动的周期关系为 $T_{a}=T_{c}>T_{b}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_472

2017 年 8 月 28 日, 中科院南极天文中心的巡天望远镜观测到一个由双中子星构成的孤立双星系统产生的引力波。该双星系统以引力波的形式向外辐射能量, 使得圆周运动的周期 $T$ 极其缓慢地减小, 双星的质量 $m_{1}$ 与 $m_{2}$ 均不变, 则下列关于该双星系统变化的说法正确的是 ( ) [图1] A: 双星间的间距逐渐增大 B: 双星间的万有引力逐渐增大 C: 双星的线速度逐渐减小 D: 双星的角速度减小

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: 2017 年 8 月 28 日, 中科院南极天文中心的巡天望远镜观测到一个由双中子星构成的孤立双星系统产生的引力波。该双星系统以引力波的形式向外辐射能量, 使得圆周运动的周期 $T$ 极其缓慢地减小, 双星的质量 $m_{1}$ 与 $m_{2}$ 均不变, 则下列关于该双星系统变化的说法正确的是 ( ) [图1] A: 双星间的间距逐渐增大 B: 双星间的万有引力逐渐增大 C: 双星的线速度逐渐减小 D: 双星的角速度减小 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

multi-modal

Astronomy_433

星体 $\mathrm{P}$ (行星或彗星) 绕太阳运动的轨迹为圆锥曲线 $r=\frac{k}{1+\varepsilon \cos \theta}$ 式中, $r$ 是 $\mathrm{P}$ 到太阳 $\mathrm{S}$ 的距离, $\theta$ 是矢径 $\mathrm{SP}$ 相对于极轴 $\mathrm{SA}$ 的夹角 (以逆时针方向为正), $k=\frac{L^{2}}{G M m^{2}}, L$是 $\mathrm{P}$ 相对于太阳的角动量, $G=6.67 \times 10^{-11} \mathrm{~m} 3 \cdot \mathrm{kg}^{-1} \cdot \mathrm{s}^{-2}$ 为引力常量, $M \approx 1.99 \times 10^{30} \mathrm{~kg}$ 为太阳的质量, $\varepsilon=\sqrt{1+\frac{2 E L^{2}}{G^{2} M^{2} m^{3}}}$ 为偏心率, $m$ 和 $E$ 分别为 $\mathrm{P}$ 的质量和机械能。假设有一颗彗星绕太阳运动的轨道为抛物线, 地球绕太阳运动的轨道可近似为圆, 两轨道相交于 $C 、 D$两点, 如图所示。已知地球轨道半径 $R_{E}=1.49 \times 10^{11} \mathrm{~m}$, 彗星轨道近日点 $A$ 到太阳的距离为地球轨道半径的三分之一, 不考虑地球和彗星之间的相互影响。求彗星经过 $C 、 D$ 两点时速度的大小 已知积分公式 $\int \frac{x d x}{\sqrt{x+a}}=\frac{2}{3}(x+a)^{\frac{3}{2}}-2 a(x+a)^{\frac{1}{2}}+C$, 式中 $\mathrm{C}$ 是任意常数。 [图1]

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题包含多个待求解的量。 问题: 星体 $\mathrm{P}$ (行星或彗星) 绕太阳运动的轨迹为圆锥曲线 $r=\frac{k}{1+\varepsilon \cos \theta}$ 式中, $r$ 是 $\mathrm{P}$ 到太阳 $\mathrm{S}$ 的距离, $\theta$ 是矢径 $\mathrm{SP}$ 相对于极轴 $\mathrm{SA}$ 的夹角 (以逆时针方向为正), $k=\frac{L^{2}}{G M m^{2}}, L$是 $\mathrm{P}$ 相对于太阳的角动量, $G=6.67 \times 10^{-11} \mathrm{~m} 3 \cdot \mathrm{kg}^{-1} \cdot \mathrm{s}^{-2}$ 为引力常量, $M \approx 1.99 \times 10^{30} \mathrm{~kg}$ 为太阳的质量, $\varepsilon=\sqrt{1+\frac{2 E L^{2}}{G^{2} M^{2} m^{3}}}$ 为偏心率, $m$ 和 $E$ 分别为 $\mathrm{P}$ 的质量和机械能。假设有一颗彗星绕太阳运动的轨道为抛物线, 地球绕太阳运动的轨道可近似为圆, 两轨道相交于 $C 、 D$两点, 如图所示。已知地球轨道半径 $R_{E}=1.49 \times 10^{11} \mathrm{~m}$, 彗星轨道近日点 $A$ 到太阳的距离为地球轨道半径的三分之一, 不考虑地球和彗星之间的相互影响。求彗星经过 $C 、 D$ 两点时速度的大小 已知积分公式 $\int \frac{x d x}{\sqrt{x+a}}=\frac{2}{3}(x+a)^{\frac{3}{2}}-2 a(x+a)^{\frac{1}{2}}+C$, 式中 $\mathrm{C}$ 是任意常数。 [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。 你的最终解答的量应该按以下顺序输出：[彗星经过 C点时速度的大小, 彗星经过 D点时速度的大小] 它们的单位依次是[m/s, m/s]，但在你给出最终答案时不应包含单位。 它们的答案类型依次是[数值, 数值] 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应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5

MPV

Astronomy

ZH

multi-modal

Astronomy_1022

The James Webb Space Telescope (JWST) is an exciting new space-based observatory which is capable of detecting incredibly faint objects that have never been seen before, but it is also possible to be seen from Earth if you have a large enough telescope. It has now entered a halo orbit around the second Lagrangian point, $L_{2}$, of the Sun-Earth system at a distance of about 1.5 million $\mathrm{km}$ from Earth, directly along the Sun-Earth line. [figure1] Figure 5: Left: An image of NASA's James Webb Space Telescope reaching its final distance from Earth. It is a tiny speck among a sea of background stars. The stars appear smudged because the telescope was tracking the motion of JWST, which appears as a small white speck. Credit: Gianluca Masi / The Virtual Telescope Project. Right: The position of the second Lagrangian point, $L_{2}$, relative to the Earth. Credit: ESA. The rectangular sunshield is rather large (measuring $21 \mathrm{~m}$ by $14 \mathrm{~m}$, roughly the same as a tennis court), very reflective (reflecting $\sim 90 \%$ of the incident light), and always points directly towards the Sun to protect the other parts of the telescope, especially to keep it cool enough to do infrared astronomy. On a very dark night, your eye's pupil opens up to about $6 \mathrm{~mm}$ in diameter and you are able to see stars as faint as $m=+6$. Estimate what diameter of telescope you would need to look through in order to see the JWST in similarly dark conditions.

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: The James Webb Space Telescope (JWST) is an exciting new space-based observatory which is capable of detecting incredibly faint objects that have never been seen before, but it is also possible to be seen from Earth if you have a large enough telescope. It has now entered a halo orbit around the second Lagrangian point, $L_{2}$, of the Sun-Earth system at a distance of about 1.5 million $\mathrm{km}$ from Earth, directly along the Sun-Earth line. [figure1] Figure 5: Left: An image of NASA's James Webb Space Telescope reaching its final distance from Earth. It is a tiny speck among a sea of background stars. The stars appear smudged because the telescope was tracking the motion of JWST, which appears as a small white speck. Credit: Gianluca Masi / The Virtual Telescope Project. Right: The position of the second Lagrangian point, $L_{2}$, relative to the Earth. Credit: ESA. The rectangular sunshield is rather large (measuring $21 \mathrm{~m}$ by $14 \mathrm{~m}$, roughly the same as a tennis court), very reflective (reflecting $\sim 90 \%$ of the incident light), and always points directly towards the Sun to protect the other parts of the telescope, especially to keep it cool enough to do infrared astronomy. On a very dark night, your eye's pupil opens up to about $6 \mathrm{~mm}$ in diameter and you are able to see stars as faint as $m=+6$. Estimate what diameter of telescope you would need to look through in order to see the JWST in similarly dark conditions. 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 cm, 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_558

我国天文学家通过 FAST, 在武仙座球状星团 $\mathrm{M}_{1} 3$ 中发现一个脉冲双星系统。如图所示, 假设在太空中有恒星 $A 、 B$ 双星系统绕点 $O$ 做顺时针匀速圆周运动, 运动周期为 $T_{1}$, 它们的轨道半径分别为 $R_{A} 、 R_{B}, R_{A}<R_{B}, C$ 为 $B$ 的卫星, 绕 $B$ 做逆时针匀速圆周运动, 周期为 $T_{2}$ 。忽略 $A$ 与 $C$ 之间的引力, $A$ 与 $B$ 之间的引力远大于 $C$ 与 $B$ 之间的引力。万有引力常量为 $G$, 求:若 $A$ 也有一颗周期为 $T_{2}$ 的卫星 $D$, 求卫星 $C 、 D$ 的轨道半径之比。 [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: 我国天文学家通过 FAST, 在武仙座球状星团 $\mathrm{M}_{1} 3$ 中发现一个脉冲双星系统。如图所示, 假设在太空中有恒星 $A 、 B$ 双星系统绕点 $O$ 做顺时针匀速圆周运动, 运动周期为 $T_{1}$, 它们的轨道半径分别为 $R_{A} 、 R_{B}, R_{A}<R_{B}, C$ 为 $B$ 的卫星, 绕 $B$ 做逆时针匀速圆周运动, 周期为 $T_{2}$ 。忽略 $A$ 与 $C$ 之间的引力, $A$ 与 $B$ 之间的引力远大于 $C$ 与 $B$ 之间的引力。万有引力常量为 $G$, 求:若 $A$ 也有一颗周期为 $T_{2}$ 的卫星 $D$, 求卫星 $C 、 D$ 的轨道半径之比。 [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_516

在刘慈欣的科幻小说《带上她的眼睛》里演绎了这样一个故事: “落日六号”地层飞船深入地球内部进行探险, 在航行中失事后下沉到地心。已知地球可视为半径为 $R$ 、质量分布均匀的球体, 且均匀球壳对壳内质点的引力为零。若地球表面的重力加速度为 $g$, 当“落日六号”位于地面以下深 $0.5 R$ 处时, 该处的重力加速度大小为 $($ ) A: $g$ B: $\frac{g}{2}$ C: $\frac{g}{4}$ D: $\frac{g}{8}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 在刘慈欣的科幻小说《带上她的眼睛》里演绎了这样一个故事: “落日六号”地层飞船深入地球内部进行探险, 在航行中失事后下沉到地心。已知地球可视为半径为 $R$ 、质量分布均匀的球体, 且均匀球壳对壳内质点的引力为零。若地球表面的重力加速度为 $g$, 当“落日六号”位于地面以下深 $0.5 R$ 处时, 该处的重力加速度大小为 $($ ) A: $g$ B: $\frac{g}{2}$ C: $\frac{g}{4}$ D: $\frac{g}{8}$ 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_807

In 2025, the Parker Solar Probe will pass just $6.9 \times 10^{6} \mathrm{~km}$ from the Sun, becoming the closest man-made object to the Sun in history. It will make five orbits, passing close to the Sun once every 89 days, before the planned end of the mission in 2026. How fast will the Parker Solar Probe be traveling at its closest approach to the Sun? A: $38 \mathrm{~km} / \mathrm{s}$ B: $48 \mathrm{~km} / \mathrm{s}$ C: $139 \mathrm{~km} / \mathrm{s}$ D: $190 \mathrm{~km} / \mathrm{s}$ E: $196 \mathrm{~km} / \mathrm{s}$

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: In 2025, the Parker Solar Probe will pass just $6.9 \times 10^{6} \mathrm{~km}$ from the Sun, becoming the closest man-made object to the Sun in history. It will make five orbits, passing close to the Sun once every 89 days, before the planned end of the mission in 2026. How fast will the Parker Solar Probe be traveling at its closest approach to the Sun? A: $38 \mathrm{~km} / \mathrm{s}$ B: $48 \mathrm{~km} / \mathrm{s}$ C: $139 \mathrm{~km} / \mathrm{s}$ D: $190 \mathrm{~km} / \mathrm{s}$ E: $196 \mathrm{~km} / \mathrm{s}$ 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_583

下列关于物理学史的说法中正确的是 ( ) A: 在牛顿之前, 亚里士多德、伽利略、笛卡尔等人就有了对力和运动的正确认识 B: 伽利略所处的时代不具备能较精确地测量自由落体运动时间的工具 C: 牛顿若能得到月球的具体运动数据, 就能通过“地月检验”测算出地球的质量 D: 开普勒通过观测天体运动, 积累下大量的数据, 总结出行星运动三大定律

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: 下列关于物理学史的说法中正确的是 ( ) A: 在牛顿之前, 亚里士多德、伽利略、笛卡尔等人就有了对力和运动的正确认识 B: 伽利略所处的时代不具备能较精确地测量自由落体运动时间的工具 C: 牛顿若能得到月球的具体运动数据, 就能通过“地月检验”测算出地球的质量 D: 开普勒通过观测天体运动, 积累下大量的数据, 总结出行星运动三大定律 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_1162

Recent years have seen an explosion in the discovery of new exoplanets. About $85 \%$ of transiting exoplanets discovered by the NASA Kepler telescope have radii less than Neptune ( $\sim 4 R_{\oplus}$ ), meaning we are improving our understanding of what the transition between rocky Earth-size planets and gaseous Neptune-size planets looks like. Given how common these "super-Earths" and "gas dwarfs" seem to be, it was odd that we didn't have any in our own Solar System. However, Batygin \& Brown (2016) suggested that a hypothetical ninth planet (called 'Planet Nine') could explain some of the unusual properties of the orbits of objects in the Kuiper Belt. This planet is inferred to have a mass of $10 M_{\oplus}$, and so would be an example of a super-Earth. [figure1] Figure 5: A plot of planet density versus radius for 33 extrasolar planets (circles) and the planets in our solar system (diamonds). Credit: Marcy et al. (2014). Analysing exoplanets discovered by Kepler, Marcy et al. (2014) used a piecewise function to describe their planetary density data such that: $$ \begin{aligned} \text { For } R_{\mathrm{P}} \leq 1.5 R_{\oplus} & \rho & =2.32+3.18 \frac{R_{\mathrm{P}}}{R_{\oplus}}\left[\mathrm{g} \mathrm{cm}^{-3}\right] \\ \text { For } 1.5 R_{\oplus}<R_{\mathrm{P}} \leq 4.2 R_{\oplus} & \frac{M_{\mathrm{P}}}{M_{\oplus}} & =2.69\left(\frac{R_{\mathrm{P}}}{R_{\oplus}}\right)^{0.93} \end{aligned} $$ where $R_{\mathrm{P}}$ is the radius of the planet, $M_{\mathrm{P}}$ is the mass of the planet, and the model's transition between rocky super-Earth and non-rocky gas dwarf occurs at $R_{\mathrm{P}}=1.5 R_{\oplus}$. The minimum speed necessary to fully escape a planet's gravity (rather than be put into an elliptical orbit) is called the escape velocity and is calculated as $$ v_{\mathrm{esc}}=\sqrt{\frac{2 G M_{\mathrm{P}}}{R_{\mathrm{P}}}} $$ where $G$ is the universal gravitational constant. In contrast, the maximum speed a rocket can provide is determined by the ejection velocity, $v_{\mathrm{e}}$, of the gas used, as determined by the chemical energy stored in the bonds of the fuel used, and the fraction of the rocket that is fuel. Since the rocket gets lighter as it burns its fuel the final speed can be bigger than $v_{\mathrm{e}}$. The rocket equation is $$ v_{\max }=v_{\mathrm{e}} \ln \frac{m_{0}}{m_{1}} $$ where $m_{0}$ is the mass at launch and $m_{1}$ is the mass once all the fuel has been burnt. The most energetic chemical reaction we can use in a rocket is hydrogen-oxygen, which gives $v_{\mathrm{e}}=4.46 \mathrm{~km} \mathrm{~s}^{-1}$, and engineering limits us to a rocket design with a maximum of $96 \%$ of launch mass being fuel (as was used with the solid rockets that launched the space shuttle).e. Calculate the maximum value of $R_{p}$ (for a rocky exoplanet) above which any alien civilization would be unable to escape their planet's gravity using simple chemical rocket propulsion systems.

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: Recent years have seen an explosion in the discovery of new exoplanets. About $85 \%$ of transiting exoplanets discovered by the NASA Kepler telescope have radii less than Neptune ( $\sim 4 R_{\oplus}$ ), meaning we are improving our understanding of what the transition between rocky Earth-size planets and gaseous Neptune-size planets looks like. Given how common these "super-Earths" and "gas dwarfs" seem to be, it was odd that we didn't have any in our own Solar System. However, Batygin \& Brown (2016) suggested that a hypothetical ninth planet (called 'Planet Nine') could explain some of the unusual properties of the orbits of objects in the Kuiper Belt. This planet is inferred to have a mass of $10 M_{\oplus}$, and so would be an example of a super-Earth. [figure1] Figure 5: A plot of planet density versus radius for 33 extrasolar planets (circles) and the planets in our solar system (diamonds). Credit: Marcy et al. (2014). Analysing exoplanets discovered by Kepler, Marcy et al. (2014) used a piecewise function to describe their planetary density data such that: $$ \begin{aligned} \text { For } R_{\mathrm{P}} \leq 1.5 R_{\oplus} & \rho & =2.32+3.18 \frac{R_{\mathrm{P}}}{R_{\oplus}}\left[\mathrm{g} \mathrm{cm}^{-3}\right] \\ \text { For } 1.5 R_{\oplus}<R_{\mathrm{P}} \leq 4.2 R_{\oplus} & \frac{M_{\mathrm{P}}}{M_{\oplus}} & =2.69\left(\frac{R_{\mathrm{P}}}{R_{\oplus}}\right)^{0.93} \end{aligned} $$ where $R_{\mathrm{P}}$ is the radius of the planet, $M_{\mathrm{P}}$ is the mass of the planet, and the model's transition between rocky super-Earth and non-rocky gas dwarf occurs at $R_{\mathrm{P}}=1.5 R_{\oplus}$. The minimum speed necessary to fully escape a planet's gravity (rather than be put into an elliptical orbit) is called the escape velocity and is calculated as $$ v_{\mathrm{esc}}=\sqrt{\frac{2 G M_{\mathrm{P}}}{R_{\mathrm{P}}}} $$ where $G$ is the universal gravitational constant. In contrast, the maximum speed a rocket can provide is determined by the ejection velocity, $v_{\mathrm{e}}$, of the gas used, as determined by the chemical energy stored in the bonds of the fuel used, and the fraction of the rocket that is fuel. Since the rocket gets lighter as it burns its fuel the final speed can be bigger than $v_{\mathrm{e}}$. The rocket equation is $$ v_{\max }=v_{\mathrm{e}} \ln \frac{m_{0}}{m_{1}} $$ where $m_{0}$ is the mass at launch and $m_{1}$ is the mass once all the fuel has been burnt. The most energetic chemical reaction we can use in a rocket is hydrogen-oxygen, which gives $v_{\mathrm{e}}=4.46 \mathrm{~km} \mathrm{~s}^{-1}$, and engineering limits us to a rocket design with a maximum of $96 \%$ of launch mass being fuel (as was used with the solid rockets that launched the space shuttle). problem: e. Calculate the maximum value of $R_{p}$ (for a rocky exoplanet) above which any alien civilization would be unable to escape their planet's gravity using simple chemical rocket propulsion systems. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of m, 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_777

Which of the following acronyms refers to an instrument of the JWST? A: NIRSpec B: HELIOS C: Exo-FMS D: HAZMAT

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 of the following acronyms refers to an instrument of the JWST? A: NIRSpec B: HELIOS C: Exo-FMS D: HAZMAT 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_954

You travel 100 miles South, 100 miles East and 100 miles North and arrive back where you started. Where are you? You are NOT at the North Pole. A: South Pole B: 100 Miles from the North Pole C: 116 Miles from the South Pole D: 200 Miles from the South Pole

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: You travel 100 miles South, 100 miles East and 100 miles North and arrive back where you started. Where are you? You are NOT at the North Pole. A: South Pole B: 100 Miles from the North Pole C: 116 Miles from the South Pole D: 200 Miles from the South Pole 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_477

如图所示, $\mathrm{A}$ 是静止在赤道上的物体, 地球自转而做匀速圆周运动。 $\mathrm{B} 、 \mathrm{C}$ 是同一平面内两颗人造卫星, $\mathrm{B}$ 位于离地高度等于地球半径的圆形轨道上, $\mathrm{C}$ 是地球同步卫星。 已知第一宇宙速度为 $v$, 物体 $\mathrm{A}$ 和卫星 $\mathrm{B} 、 \mathrm{C}$ 的线速度大小分别为 $v_{A} 、 v_{B} 、 v_{C}$, 运动周期大小分别为 $T_{A} 、 T_{B} 、 T_{C}$, 下列关系正确的是（） [图1] A: $T_{A}=T_{C}<T_{B}$ B: $T_{A}=T_{C}>T_{B}$ C: $v_{A}<v_{C}<v_{B}<v$ D: $v_{A}<v_{B}<v_{C}<v$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 如图所示, $\mathrm{A}$ 是静止在赤道上的物体, 地球自转而做匀速圆周运动。 $\mathrm{B} 、 \mathrm{C}$ 是同一平面内两颗人造卫星, $\mathrm{B}$ 位于离地高度等于地球半径的圆形轨道上, $\mathrm{C}$ 是地球同步卫星。 已知第一宇宙速度为 $v$, 物体 $\mathrm{A}$ 和卫星 $\mathrm{B} 、 \mathrm{C}$ 的线速度大小分别为 $v_{A} 、 v_{B} 、 v_{C}$, 运动周期大小分别为 $T_{A} 、 T_{B} 、 T_{C}$, 下列关系正确的是（） [图1] A: $T_{A}=T_{C}<T_{B}$ B: $T_{A}=T_{C}>T_{B}$ C: $v_{A}<v_{C}<v_{B}<v$ D: $v_{A}<v_{B}<v_{C}<v$ 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应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_986

Earth's moon has a radius of $1737 \mathrm{~km}$ and Titan (one of Saturn's moons) has a radius of $2576 \mathrm{~km}$. At the surface, their gravitational field strengths, $g$, are $1.63 \mathrm{~N} \mathrm{~kg}^{-1}$ and $1.35 \mathrm{~N} \mathrm{~kg}^{-1}$ respectively. Determine the ratio of masses, $M_{\text {Titan }} / M_{\text {Moon }}$, given $g \propto M / R^{2}$. A: 1.23 B: 1.79 C: 1.82 D: 2.66

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: Earth's moon has a radius of $1737 \mathrm{~km}$ and Titan (one of Saturn's moons) has a radius of $2576 \mathrm{~km}$. At the surface, their gravitational field strengths, $g$, are $1.63 \mathrm{~N} \mathrm{~kg}^{-1}$ and $1.35 \mathrm{~N} \mathrm{~kg}^{-1}$ respectively. Determine the ratio of masses, $M_{\text {Titan }} / M_{\text {Moon }}$, given $g \propto M / R^{2}$. A: 1.23 B: 1.79 C: 1.82 D: 2.66 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_591

在地球上通过望远镜观察某种双星, 视线与双星轨道共面。观测发现每隔时间 $T$ 两颗恒星与望远镜共线一次, 已知两颗恒星 $\mathrm{A} 、 \mathrm{~B}$ 间距为 $d$, 万有引力常量为 $G$, 则可推算出双星系统的总质量为 ( ) [图1] A: $\frac{\pi^{2} d^{2}}{G T^{2}}$ B: $\frac{\pi^{2} d^{3}}{G T^{2}}$ C: $\frac{2 \pi^{2} d^{2}}{G T^{2}}$ D: $\frac{4 \pi^{2} d^{2}}{G T^{2}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 在地球上通过望远镜观察某种双星, 视线与双星轨道共面。观测发现每隔时间 $T$ 两颗恒星与望远镜共线一次, 已知两颗恒星 $\mathrm{A} 、 \mathrm{~B}$ 间距为 $d$, 万有引力常量为 $G$, 则可推算出双星系统的总质量为 ( ) [图1] A: $\frac{\pi^{2} d^{2}}{G T^{2}}$ B: $\frac{\pi^{2} d^{3}}{G T^{2}}$ C: $\frac{2 \pi^{2} d^{2}}{G T^{2}}$ D: $\frac{4 \pi^{2} d^{2}}{G T^{2}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

multi-modal

Astronomy_47

2019 年 1 月 3 日, 我国研制的嫦娥四号探测器成功降落在月球背面, 实现了人类历史上与月球背面的第一次“亲密接触”. 2020 年前将实现“回”的任务, 即飞行器不但在月球上落下来, 还要取一些东西带回地球, 并计划在 2030 年前后实现航天员登月. 若某航天员分别在月球和地球表面离地高度 $h$ 处, 以相同的初速度水平抛出物体, 在地球上物体的抛出点与落地点间的距离为 $\sqrt{2} h$. 已知地球质量为月球质量的 81 倍, 地球半径为月球半径的 4 倍, 地球表面的重力加速度为 $\mathrm{g}$, 忽略空气阻力. 下列说法正确的是 A: 嫦娥四号的发射速度必须大于第二宇宙速度 B: 嫦娥四号在月球背面着陆过程中如果关闭发动机, 其加速度为 $\frac{4}{9} g$ C: 嫦娥四号在月球背面着陆过程中如果关闭发动机, 其内部物体处于完全失重状态 D: 在月球上抛出的物体的抛出点与落地点间的距离为 $\frac{\sqrt{97}}{4} h$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 2019 年 1 月 3 日, 我国研制的嫦娥四号探测器成功降落在月球背面, 实现了人类历史上与月球背面的第一次“亲密接触”. 2020 年前将实现“回”的任务, 即飞行器不但在月球上落下来, 还要取一些东西带回地球, 并计划在 2030 年前后实现航天员登月. 若某航天员分别在月球和地球表面离地高度 $h$ 处, 以相同的初速度水平抛出物体, 在地球上物体的抛出点与落地点间的距离为 $\sqrt{2} h$. 已知地球质量为月球质量的 81 倍, 地球半径为月球半径的 4 倍, 地球表面的重力加速度为 $\mathrm{g}$, 忽略空气阻力. 下列说法正确的是 A: 嫦娥四号的发射速度必须大于第二宇宙速度 B: 嫦娥四号在月球背面着陆过程中如果关闭发动机, 其加速度为 $\frac{4}{9} g$ C: 嫦娥四号在月球背面着陆过程中如果关闭发动机, 其内部物体处于完全失重状态 D: 在月球上抛出的物体的抛出点与落地点间的距离为 $\frac{\sqrt{97}}{4} h$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

text-only

Astronomy_814

A comet called "LQi2017" which has physical values of $e=1.2, a=19 \mathrm{AU}$ was visible from the Earth in 2017. In which year is this comet visible again? A: 2080 B: 2100 C: 2109 D: 2130 E: The comet is not visible again.

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 called "LQi2017" which has physical values of $e=1.2, a=19 \mathrm{AU}$ was visible from the Earth in 2017. In which year is this comet visible again? A: 2080 B: 2100 C: 2109 D: 2130 E: The comet is not visible again. 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

EN

text-only

Astronomy_1079

Wolf-Rayet (WR) stars are some of the hottest stars known, with very strong stellar winds causing considerable mass to the be lost to the interstellar medium (ISM). In binary systems between a WR star and a very large $\mathrm{O}$ or $\mathrm{B}$ spectral class star, where their strong stellar winds collide can create the conditions for the formation of dust which goes on to enrich the ISM. [figure1] Figure 3: Left: A view of the WR140 binary system taken with the James Webb Space Telescope (JWST) in July 2022, showing clearly at least 17 nested dust shells. Credit: NASA/ESA/CSA/STScI/JPL-Caltech. Right: A radial plot along the image in the three mid infrared JWST filters used corresponding to 7.7, 15 and $21 \mu \mathrm{m}$, as well as the model of the dust production. The peaks correspond to each shell. Shells 2 and 17 are indicated on the model and the median shell separation is shown by the grey vertical lines. The projected distance is given in arcseconds. Credit: Lau et al. (2022). The WR140 system consists of a WR and an O star which produce dust very regularly when the two stars are close together, around periastron. They are in a highly elliptical orbit $(e=0.8993)$ with a period of 2895 days. Once far from the stars, these dust shells move through space at a remarkably constant speed as indicated by the regularity of the shells in the recent image taken with the James Webb Space Telescope (JWST), shown above in Figure 3. An artist's impression of the two stars in the system and the orbit (in the reference frame of the WR star) is shown in Figure 4 below. [figure2] Figure 4: Left: The relative size of the Sun, upper left, compared to the two stars in the system WR140. The O-type star is $\sim 30 M_{\odot}$, while its companion is $\sim 10 M_{\odot}$. Credit: NASA/JPL-Caltech. Right: The projected orbital configuration of WR 140 in the reference frame of the WR star. The red solid region around the periastron passage is where the O star is when dust is being formed. Credit: Lau et al. (2022).a. Take the distance to the system to be $1.64 \mathrm{kpc}$. (i) By taking measurements from Figure 3, show that the average radial expansion velocity between shells 2 to 17 is $\sim 2600 \mathrm{~km} \mathrm{~s}^{-1}$. [1 arcsecond $=1^{\prime \prime}$ is a measurement of angle, where $3600^{\prime \prime}=1^{\circ}$.]

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: Wolf-Rayet (WR) stars are some of the hottest stars known, with very strong stellar winds causing considerable mass to the be lost to the interstellar medium (ISM). In binary systems between a WR star and a very large $\mathrm{O}$ or $\mathrm{B}$ spectral class star, where their strong stellar winds collide can create the conditions for the formation of dust which goes on to enrich the ISM. [figure1] Figure 3: Left: A view of the WR140 binary system taken with the James Webb Space Telescope (JWST) in July 2022, showing clearly at least 17 nested dust shells. Credit: NASA/ESA/CSA/STScI/JPL-Caltech. Right: A radial plot along the image in the three mid infrared JWST filters used corresponding to 7.7, 15 and $21 \mu \mathrm{m}$, as well as the model of the dust production. The peaks correspond to each shell. Shells 2 and 17 are indicated on the model and the median shell separation is shown by the grey vertical lines. The projected distance is given in arcseconds. Credit: Lau et al. (2022). The WR140 system consists of a WR and an O star which produce dust very regularly when the two stars are close together, around periastron. They are in a highly elliptical orbit $(e=0.8993)$ with a period of 2895 days. Once far from the stars, these dust shells move through space at a remarkably constant speed as indicated by the regularity of the shells in the recent image taken with the James Webb Space Telescope (JWST), shown above in Figure 3. An artist's impression of the two stars in the system and the orbit (in the reference frame of the WR star) is shown in Figure 4 below. [figure2] Figure 4: Left: The relative size of the Sun, upper left, compared to the two stars in the system WR140. The O-type star is $\sim 30 M_{\odot}$, while its companion is $\sim 10 M_{\odot}$. Credit: NASA/JPL-Caltech. Right: The projected orbital configuration of WR 140 in the reference frame of the WR star. The red solid region around the periastron passage is where the O star is when dust is being formed. Credit: Lau et al. (2022). problem: a. Take the distance to the system to be $1.64 \mathrm{kpc}$. (i) By taking measurements from Figure 3, show that the average radial expansion velocity between shells 2 to 17 is $\sim 2600 \mathrm{~km} \mathrm{~s}^{-1}$. [1 arcsecond $=1^{\prime \prime}$ is a measurement of angle, where $3600^{\prime \prime}=1^{\circ}$.] All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of \mathrm{~km} \mathrm{~s}^{-1}, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

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Astronomy

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multi-modal

Astronomy_674

马斯克的 SpaceX“猎鹰”重型火箭将一辆跑车发射到太空, 其轨道示意图如图中椭圆 II 所示, 其中 $A 、 C$ 分别是近日点和远日点, 图中 I、III 轨道分别为地球和火星绕太阳运动的圆轨道, $B$ 点为轨道 II、III 的交点, 若运动中只考虑太阳的万有引力, 则以下说法正确的是（ ） [图1] A: 跑车经过 $A$ 点时的速率大于火星绕日的速率 B: 跑车经过 $B$ 点时的加速度等于火星经过 $B$ 点时的加速度 C: 跑车在 $C$ 点的速率一定大于火星绕日的速率 D: 跑车在 $C$ 点的速率可能等于火星绕日的速率

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: 马斯克的 SpaceX“猎鹰”重型火箭将一辆跑车发射到太空, 其轨道示意图如图中椭圆 II 所示, 其中 $A 、 C$ 分别是近日点和远日点, 图中 I、III 轨道分别为地球和火星绕太阳运动的圆轨道, $B$ 点为轨道 II、III 的交点, 若运动中只考虑太阳的万有引力, 则以下说法正确的是（ ） [图1] A: 跑车经过 $A$ 点时的速率大于火星绕日的速率 B: 跑车经过 $B$ 点时的加速度等于火星经过 $B$ 点时的加速度 C: 跑车在 $C$ 点的速率一定大于火星绕日的速率 D: 跑车在 $C$ 点的速率可能等于火星绕日的速率 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_741

What is the so-called bolometric luminosity in astronomy? A: The luminosity, integrated over vertically polarized wavelengths. B: The luminosity, integrated over horizontally wavelengths. C: The luminosity, integrated over visible wavelengths. D: The luminosity, integrated over all wavelengths.

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 so-called bolometric luminosity in astronomy? A: The luminosity, integrated over vertically polarized wavelengths. B: The luminosity, integrated over horizontally wavelengths. C: The luminosity, integrated over visible wavelengths. D: The luminosity, integrated over all wavelengths. 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_881

The exoplanet HD 209458b has a mass of 0.71 Jupiter masses and orbits HD 209458 with an orbital period of 3.53 days. HD 209458 has a mass of 1.15 Solar masses. Assuming that the orbit of HD 209458b is circular (which is a good approximation here) and that its orbit lies perfectly in our line of sight, what is the radial velocity semi-amplitude of HD 209458 due to the orbital motion of HD $209458 \mathrm{~b}$, in $\mathrm{m} / \mathrm{s}$ ? A: $69.6 \mathrm{~m} / \mathrm{s}$ B: $85.9 \mathrm{~m} / \mathrm{s}$ C: $94.2 \mathrm{~m} / \mathrm{s}$ D: $120.8 \mathrm{~m} / \mathrm{s}$

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: The exoplanet HD 209458b has a mass of 0.71 Jupiter masses and orbits HD 209458 with an orbital period of 3.53 days. HD 209458 has a mass of 1.15 Solar masses. Assuming that the orbit of HD 209458b is circular (which is a good approximation here) and that its orbit lies perfectly in our line of sight, what is the radial velocity semi-amplitude of HD 209458 due to the orbital motion of HD $209458 \mathrm{~b}$, in $\mathrm{m} / \mathrm{s}$ ? A: $69.6 \mathrm{~m} / \mathrm{s}$ B: $85.9 \mathrm{~m} / \mathrm{s}$ C: $94.2 \mathrm{~m} / \mathrm{s}$ D: $120.8 \mathrm{~m} / \mathrm{s}$ You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

SC

Astronomy

EN

text-only

Astronomy_450

已知引力常量为 $G$, 星球的质量 $M$, 星球的半径 $R$, 飞船在轨道 I 上运动时的质量 $m, P 、 Q$ 点与星球表面的高度分别 $h_{1} 、 h_{2}$, 飞船与星球中心的距离为 $r$ 时, 引力势能为 $E_{\mathrm{p}}=-G \frac{M m}{r}$ (取无穷远处引力势能为零), 飞船经过 $Q$ 点的速度大小为 $v$, 在 $P$ 点由轨道 I 变为轨道 II 的过程中, 发动机沿轨道的切线方向瞬间一次性喷出一部分气体, 喷出的气体相对喷气后飞船的速度大小为 $u$ ，则下列说法正确的是（ ） [图1] A: 飞船在圆形轨道 I 上运动的速度大小约 $\sqrt{\frac{G M}{R+h_{1}}}$ B: 飞船经过 $P$ 点时的速度大小为 $\sqrt{v^{2}-2 G M\left(\frac{1}{R+h_{1}}-\frac{1}{R+h_{2}}\right)}$ C: 飞船在轨道 II 上运动时速度大小变化 D: 喷出的气体的质量为 $\frac{m}{u}\left[\sqrt{\frac{G M}{R+h_{1}}}-\sqrt{v^{2}+2 G M\left(\frac{1}{R+h_{1}}-\frac{1}{R+h_{2}}\right)}\right]$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 已知引力常量为 $G$, 星球的质量 $M$, 星球的半径 $R$, 飞船在轨道 I 上运动时的质量 $m, P 、 Q$ 点与星球表面的高度分别 $h_{1} 、 h_{2}$, 飞船与星球中心的距离为 $r$ 时, 引力势能为 $E_{\mathrm{p}}=-G \frac{M m}{r}$ (取无穷远处引力势能为零), 飞船经过 $Q$ 点的速度大小为 $v$, 在 $P$ 点由轨道 I 变为轨道 II 的过程中, 发动机沿轨道的切线方向瞬间一次性喷出一部分气体, 喷出的气体相对喷气后飞船的速度大小为 $u$ ，则下列说法正确的是（ ） [图1] A: 飞船在圆形轨道 I 上运动的速度大小约 $\sqrt{\frac{G M}{R+h_{1}}}$ B: 飞船经过 $P$ 点时的速度大小为 $\sqrt{v^{2}-2 G M\left(\frac{1}{R+h_{1}}-\frac{1}{R+h_{2}}\right)}$ C: 飞船在轨道 II 上运动时速度大小变化 D: 喷出的气体的质量为 $\frac{m}{u}\left[\sqrt{\frac{G M}{R+h_{1}}}-\sqrt{v^{2}+2 G M\left(\frac{1}{R+h_{1}}-\frac{1}{R+h_{2}}\right)}\right]$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_551

2023 年, 中国将全面推进探月工程四期, 规划包括嫦娥六号、嫦娥七号和嫦蛾八号任务。其中嫦娥七号准备在月球南极着陆, 主要任务是开展飞跃探测, 争取能找到水。假设质量为 $m$ 的嫦娥七号探测器在距离月面的高度等于月球半径处绕着月球表面做匀速圆周运动时, 其周期为 $T_{1}$, 当探测器停在月球的两极时, 测得重力加速度的大小为 $g_{0}$, 已知月球自转的周期为 $T_{2}$, 引力常量为 $G$, 月球视为均匀球体, 下列说法正确的是 ( ) 国家航天局表示 探月工程四期 今年正:式启动工程㖄制 未来将发射㲖娥六号、㗂娥七号、嫦娥八号探测器 嫦娥六号计划到月球背而采样 [图1] A: 月球的半径为 $\frac{g_{0} T_{1}^{2}}{16 \pi}$ B: 月球的第一宇宙速度为 $\frac{g_{0} T_{1}}{8 \pi}$ C: 当停在月球赤道上时, 探测器受到水平面的支持力为 $\frac{\left(8 T_{2}^{2}-T_{1}^{2}\right) m g_{0}}{8 T_{2}^{2}}$ D: 当停在月球上纬度为 $60^{\circ}$ 的区域时, 探测器随月球转动的线速度为 $\frac{g_{0} T_{1}^{2}}{16 \pi T_{2}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 2023 年, 中国将全面推进探月工程四期, 规划包括嫦娥六号、嫦娥七号和嫦蛾八号任务。其中嫦娥七号准备在月球南极着陆, 主要任务是开展飞跃探测, 争取能找到水。假设质量为 $m$ 的嫦娥七号探测器在距离月面的高度等于月球半径处绕着月球表面做匀速圆周运动时, 其周期为 $T_{1}$, 当探测器停在月球的两极时, 测得重力加速度的大小为 $g_{0}$, 已知月球自转的周期为 $T_{2}$, 引力常量为 $G$, 月球视为均匀球体, 下列说法正确的是 ( ) 国家航天局表示 探月工程四期 今年正:式启动工程㖄制 未来将发射㲖娥六号、㗂娥七号、嫦娥八号探测器 嫦娥六号计划到月球背而采样 [图1] A: 月球的半径为 $\frac{g_{0} T_{1}^{2}}{16 \pi}$ B: 月球的第一宇宙速度为 $\frac{g_{0} T_{1}}{8 \pi}$ C: 当停在月球赤道上时, 探测器受到水平面的支持力为 $\frac{\left(8 T_{2}^{2}-T_{1}^{2}\right) m g_{0}}{8 T_{2}^{2}}$ D: 当停在月球上纬度为 $60^{\circ}$ 的区域时, 探测器随月球转动的线速度为 $\frac{g_{0} T_{1}^{2}}{16 \pi T_{2}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

multi-modal

Astronomy_1183

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.d. Derive a formula for the distance, $D$, as a function of $\theta, \mu_{a}$ and $\mu_{r}$ (i.e. independent of $\beta$ ), and hence calculate $\theta$.

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: d. Derive a formula for the distance, $D$, as a function of $\theta, \mu_{a}$ and $\mu_{r}$ (i.e. independent of $\beta$ ), and hence calculate $\theta$. 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_734

航天技术的发展是当今各国综合国力的直接体现，近年来，我国的航天技术取得了让世界瞩目的成绩，也引领科技爱好者思索航天技术的发展，有人就提出了一种不用火箭发射人造地球卫星的设想。其设想如下: 如图所示, 在地球上距地心 $h$ 处沿一条弦挖一光滑通道, 在通道的两个出口处 $A$ 和 $B$ 分别将质量为 $M$ 的物体和质量为 $m$ 的待发射卫星同时自由释放, $M \gg m$, 在中点 $O^{\prime}$ 弹性正撞后，质量为 $m$ 的物体，即待发射的卫星就会从通道口 $B$ 冲出通道, 设置一个装置, 卫星从 $B$ 冲出就把速度变为沿地球切线方向, 但不改变速度大小, 这样就有可能成功发射卫星。已知地球可视为质量分布均匀 的球体, 且质量分布均匀的球壳对壳内物体的引力为零, 地球半径为 $R_{0}$, 表面的重力加速度为 $g$ 。求卫星成功发射时, $h$ 的最大值。

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: 航天技术的发展是当今各国综合国力的直接体现，近年来，我国的航天技术取得了让世界瞩目的成绩，也引领科技爱好者思索航天技术的发展，有人就提出了一种不用火箭发射人造地球卫星的设想。其设想如下: 如图所示, 在地球上距地心 $h$ 处沿一条弦挖一光滑通道, 在通道的两个出口处 $A$ 和 $B$ 分别将质量为 $M$ 的物体和质量为 $m$ 的待发射卫星同时自由释放, $M \gg m$, 在中点 $O^{\prime}$ 弹性正撞后，质量为 $m$ 的物体，即待发射的卫星就会从通道口 $B$ 冲出通道, 设置一个装置, 卫星从 $B$ 冲出就把速度变为沿地球切线方向, 但不改变速度大小, 这样就有可能成功发射卫星。已知地球可视为质量分布均匀 的球体, 且质量分布均匀的球壳对壳内物体的引力为零, 地球半径为 $R_{0}$, 表面的重力加速度为 $g$ 。求卫星成功发射时, $h$ 的最大值。 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_1159

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 $7.35 \times 10^{22} \mathrm{~kg}$.c. For the Hohmann transfer orbit (dashed line), find its semi-major axis and hence the duration of a translunar coast from A to B (expressed in hours and minutes).

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

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Astronomy_298

一颗距离地面高度等于地球半径 $R$ 的圆形轨道地球卫星, 其轨道平面与赤道平面重合。已知地球同步卫星轨道高于该卫星轨道, 地球表面重力加速度为 $g$, 则下列说法正确的是 ( ) A: 该卫星绕地球运动的周期 $T=4 \pi \sqrt{\frac{R_{0}}{g}}$ B: 该卫星的线速度小于地球同步卫星的线速度 C: 该卫星绕地球运动的加速度大小 $a=\frac{g}{2}$ D: 若该卫星绕行方向也是自西向东, 则赤道上的一个固定点连续两次经过该卫星正下方的时间间隔大于该卫星的周期

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 一颗距离地面高度等于地球半径 $R$ 的圆形轨道地球卫星, 其轨道平面与赤道平面重合。已知地球同步卫星轨道高于该卫星轨道, 地球表面重力加速度为 $g$, 则下列说法正确的是 ( ) A: 该卫星绕地球运动的周期 $T=4 \pi \sqrt{\frac{R_{0}}{g}}$ B: 该卫星的线速度小于地球同步卫星的线速度 C: 该卫星绕地球运动的加速度大小 $a=\frac{g}{2}$ D: 若该卫星绕行方向也是自西向东, 则赤道上的一个固定点连续两次经过该卫星正下方的时间间隔大于该卫星的周期 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

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text-only

Astronomy_974

'Oumuamua is an elongated interstellar comet. During an observation with a ground-based telescope whilst close to the Sun, the light curve displayed a sinusoidal variation in magnitude as shown below. If the comet is modelled as an ellipsoid with a shortest visible axis of $30 \mathrm{~m}$, use the light curve to determine the approximate size of the longest visible axis. Assume that the major axis lies in the plane along the line of sight. [figure1] A: $\sim 60 \mathrm{~m}$ B: $\sim 240 \mathrm{~m}$ C: $\sim 960 \mathrm{~m}$ D: $\sim 3840 \mathrm{~m}$

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: 'Oumuamua is an elongated interstellar comet. During an observation with a ground-based telescope whilst close to the Sun, the light curve displayed a sinusoidal variation in magnitude as shown below. If the comet is modelled as an ellipsoid with a shortest visible axis of $30 \mathrm{~m}$, use the light curve to determine the approximate size of the longest visible axis. Assume that the major axis lies in the plane along the line of sight. [figure1] A: $\sim 60 \mathrm{~m}$ B: $\sim 240 \mathrm{~m}$ C: $\sim 960 \mathrm{~m}$ D: $\sim 3840 \mathrm{~m}$ 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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multi-modal

Astronomy_976

In February 2017 the astronomical world was very excited by the announcement of the discovery of seven exoplanets similar in size to Earth orbiting around the ultra-cool red dwarf star TRAPPIST-1, found only $12.14 \mathrm{pc}$ away. Whilst they are much closer to their star than Mercury is to the Sun, since the star is considerably cooler it means that three of them (planets d, e and f) are in the habitable zone of the star (defined as where liquid water could exist on the surface). [figure1] Figure 4: A size comparison of the planets of the TRAPPIST-1 system, lined up in order of increasing distance from their host star. The planetary surfaces are portrayed with an artist's impression of their potential surface features, including water, ice, and atmospheres. Credit: NASA/R. Hurt/T. Pyle Data about the system is below: | | Mass / $M_{E}$ | Radius / $R_{E}$ | Orbital Period / Earth days | | :---: | :---: | :---: | :---: | | TRAPPIST-1b | 1.02 | 1.127 | 1.511 | | TRAPPIST-1c | 1.16 | 1.100 | 2.422 | | TRAPPIST-1d | 0.30 | 0.788 | 4.050 | | TRAPPIST-1e | 0.77 | 0.915 | 6.099 | | TRAPPIST-1f | 0.93 | 1.052 | 9.206 | | TRAPPIST-1g | 1.15 | 1.154 | 12.354 | | TRAPPIST-1h | 0.33 | 0.777 | 18.768 | The star has a mass of $0.089 M_{\odot}$, radius $0.121 R_{\odot}$, luminosity $5.22 \times 10^{-4} L_{\odot}$ and effective surface temperature of $2511 \mathrm{~K}$. Assume the planets orbit in circular, coplanar orbits. For black-body radiation (which is a good approximation for a stellar spectrum), the modal wavelength in the spectrum (in terms of intensity), $\lambda_{\max }$, is related to the effective temperature, $T$, through Wien's displacement law, $$ \lambda_{\max } T=2.90 \times 10^{-3} \mathrm{~m} \mathrm{~K} $$ The smallest angular width, $\theta_{\min }$, in radians that a telescope can resolve is limited by diffraction (ignoring the effects of the atmosphere). For a telescope of aperture diameter $D$, operating at wavelength $\lambda$, it can be shown that $$ \theta_{\min }=\frac{1.22 \lambda}{D} . $$ Assuming life developed on both planets $\mathrm{d}$ and $\mathrm{f}$, calculate the size of telescope an alien on planet $\mathrm{f}$ would need to be able to resolve a city $20 \mathrm{~km}$ across on planet $\mathrm{d}$ when they are at their closest in their orbits. Is such a telescope feasible? [Assume their eyes have evolved to operate around the modal wavelength of the star.] [Hint: You may find the small angle approximation useful: $\tan x \approx x$ when $|x| \ll 1$ and in radians. A radian is a measure of angle, and there are $2 \pi$ radians in a circle (i.e. $2 \pi$ radians $=360^{\circ}$ ).]

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. problem: In February 2017 the astronomical world was very excited by the announcement of the discovery of seven exoplanets similar in size to Earth orbiting around the ultra-cool red dwarf star TRAPPIST-1, found only $12.14 \mathrm{pc}$ away. Whilst they are much closer to their star than Mercury is to the Sun, since the star is considerably cooler it means that three of them (planets d, e and f) are in the habitable zone of the star (defined as where liquid water could exist on the surface). [figure1] Figure 4: A size comparison of the planets of the TRAPPIST-1 system, lined up in order of increasing distance from their host star. The planetary surfaces are portrayed with an artist's impression of their potential surface features, including water, ice, and atmospheres. Credit: NASA/R. Hurt/T. Pyle Data about the system is below: | | Mass / $M_{E}$ | Radius / $R_{E}$ | Orbital Period / Earth days | | :---: | :---: | :---: | :---: | | TRAPPIST-1b | 1.02 | 1.127 | 1.511 | | TRAPPIST-1c | 1.16 | 1.100 | 2.422 | | TRAPPIST-1d | 0.30 | 0.788 | 4.050 | | TRAPPIST-1e | 0.77 | 0.915 | 6.099 | | TRAPPIST-1f | 0.93 | 1.052 | 9.206 | | TRAPPIST-1g | 1.15 | 1.154 | 12.354 | | TRAPPIST-1h | 0.33 | 0.777 | 18.768 | The star has a mass of $0.089 M_{\odot}$, radius $0.121 R_{\odot}$, luminosity $5.22 \times 10^{-4} L_{\odot}$ and effective surface temperature of $2511 \mathrm{~K}$. Assume the planets orbit in circular, coplanar orbits. For black-body radiation (which is a good approximation for a stellar spectrum), the modal wavelength in the spectrum (in terms of intensity), $\lambda_{\max }$, is related to the effective temperature, $T$, through Wien's displacement law, $$ \lambda_{\max } T=2.90 \times 10^{-3} \mathrm{~m} \mathrm{~K} $$ The smallest angular width, $\theta_{\min }$, in radians that a telescope can resolve is limited by diffraction (ignoring the effects of the atmosphere). For a telescope of aperture diameter $D$, operating at wavelength $\lambda$, it can be shown that $$ \theta_{\min }=\frac{1.22 \lambda}{D} . $$ Assuming life developed on both planets $\mathrm{d}$ and $\mathrm{f}$, calculate the size of telescope an alien on planet $\mathrm{f}$ would need to be able to resolve a city $20 \mathrm{~km}$ across on planet $\mathrm{d}$ when they are at their closest in their orbits. Is such a telescope feasible? [Assume their eyes have evolved to operate around the modal wavelength of the star.] [Hint: You may find the small angle approximation useful: $\tan x \approx x$ when $|x| \ll 1$ and in radians. A radian is a measure of angle, and there are $2 \pi$ radians in a circle (i.e. $2 \pi$ radians $=360^{\circ}$ ).] 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_799

Consider a $\mathrm{f} / 9$ telescope with focal length $f=1.0 \mathrm{~m}$ that operates at visible wavelength $\lambda=$ $5000 \AA$. What is the farthest distance at which an open cluster of radius $R_{C}=4.1 \mathrm{pc}$ can be resolved by this telescope? A: $1.2 \times 10^{6} \mathrm{pc}$ B: $1.5 \times 10^{6} \mathrm{pc}$ C: $3.0 \times 10^{6} \mathrm{pc}$ D: $4.2 \times 10^{6} \mathrm{pc}$ E: $5.8 \times 10^{6} \mathrm{pc}$

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: Consider a $\mathrm{f} / 9$ telescope with focal length $f=1.0 \mathrm{~m}$ that operates at visible wavelength $\lambda=$ $5000 \AA$. What is the farthest distance at which an open cluster of radius $R_{C}=4.1 \mathrm{pc}$ can be resolved by this telescope? A: $1.2 \times 10^{6} \mathrm{pc}$ B: $1.5 \times 10^{6} \mathrm{pc}$ C: $3.0 \times 10^{6} \mathrm{pc}$ D: $4.2 \times 10^{6} \mathrm{pc}$ E: $5.8 \times 10^{6} \mathrm{pc}$ 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

EN

text-only

Astronomy_970

Main sequence stars fuse hydrogen atoms to form helium in their cores. About $90 \%$ of the stars in the Universe, including the Sun, are main sequence stars. These stars can range from about a tenth of the mass of the Sun to up to 200 times as massive. The main source of energy in main sequence stars is from nuclear fusion. The mass of one hydrogen nucleus is $m_{\mathrm{H}}=1.674 \times 10^{-27} \mathrm{~kg}$, and the mass of one helium nucleus is $m_{\mathrm{He}}=6.649 \times 10^{-27} \mathrm{~kg}$. [figure1] Figure 3: Left: The proton-proton chain reaction is one of two known sets of nuclear fusion reactions by which stars convert hydrogen to helium. It dominates in stars with masses less than or equal to that of the Sun's, and involves a net change of four hydrogen nuclei becoming one helium nucleus. Right: Only the core of a main sequence star will undergo nuclear fusion due to the higher temperature than the surrounding hydrogen shell. For main sequence stars the luminosity scales with the mass as $L \propto M^{3.5}$. Consider a binary star system with two stars separated by 9.1 au and an orbital period of 3.83 years. If the stars are unequal in mass, with one being three times heavier than the other, and assuming similar structures to the Sun, estimate the main sequence lifetime of the larger star.

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: Main sequence stars fuse hydrogen atoms to form helium in their cores. About $90 \%$ of the stars in the Universe, including the Sun, are main sequence stars. These stars can range from about a tenth of the mass of the Sun to up to 200 times as massive. The main source of energy in main sequence stars is from nuclear fusion. The mass of one hydrogen nucleus is $m_{\mathrm{H}}=1.674 \times 10^{-27} \mathrm{~kg}$, and the mass of one helium nucleus is $m_{\mathrm{He}}=6.649 \times 10^{-27} \mathrm{~kg}$. [figure1] Figure 3: Left: The proton-proton chain reaction is one of two known sets of nuclear fusion reactions by which stars convert hydrogen to helium. It dominates in stars with masses less than or equal to that of the Sun's, and involves a net change of four hydrogen nuclei becoming one helium nucleus. Right: Only the core of a main sequence star will undergo nuclear fusion due to the higher temperature than the surrounding hydrogen shell. For main sequence stars the luminosity scales with the mass as $L \propto M^{3.5}$. Consider a binary star system with two stars separated by 9.1 au and an orbital period of 3.83 years. If the stars are unequal in mass, with one being three times heavier than the other, and assuming similar structures to the Sun, estimate the main sequence lifetime of the larger star. 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.

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Astronomy

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multi-modal

Astronomy_742

Which one of these wavelengths is considered A: 150 meters B: 150 millimeters C: 150 micrometers D: 150 nanometers

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 these wavelengths is considered A: 150 meters B: 150 millimeters C: 150 micrometers D: 150 nanometers 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

EN

text-only

Astronomy_464

某行星的自转周期为 $T$, 赤道半径为 $R$ 。研究发现, 当该行星的自转角速度变为原来的 2 倍时会导致该行星赤道上的物体恰好对行星表面没有压力。已知引力常量为 $G$, 则 A: 该行星的质量 $M=\frac{4 \pi R^{3}}{G \mathrm{~T}^{2}}$ B: 该行星的同步卫星的轨道半径 $r=\sqrt[3]{4} R$ C: 质量为 $m$ 的物体对该行星赤道表面的压力 $F=\frac{16 \pi^{2} m R}{\mathrm{~T}^{2}}$ D: 环绕该行星做匀速圆周运动的卫星的最大线速度为 $7.9 \mathrm{~km} / \mathrm{s}$

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: 某行星的自转周期为 $T$, 赤道半径为 $R$ 。研究发现, 当该行星的自转角速度变为原来的 2 倍时会导致该行星赤道上的物体恰好对行星表面没有压力。已知引力常量为 $G$, 则 A: 该行星的质量 $M=\frac{4 \pi R^{3}}{G \mathrm{~T}^{2}}$ B: 该行星的同步卫星的轨道半径 $r=\sqrt[3]{4} R$ C: 质量为 $m$ 的物体对该行星赤道表面的压力 $F=\frac{16 \pi^{2} m R}{\mathrm{~T}^{2}}$ D: 环绕该行星做匀速圆周运动的卫星的最大线速度为 $7.9 \mathrm{~km} / \mathrm{s}$ 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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text-only

Astronomy_847

A predicted (A) and observed (B) rotation curve of a typical spiral galaxy is shown above. What component of the galaxy causes this discrepancy? A: Baryons B: Neutrinos C: Gamma Rays D: Dark Matter E: Globular Clusters

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 predicted (A) and observed (B) rotation curve of a typical spiral galaxy is shown above. What component of the galaxy causes this discrepancy? A: Baryons B: Neutrinos C: Gamma Rays D: Dark Matter E: Globular Clusters 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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text-only

Astronomy_400

《流浪地球 2 》中的太空电梯令人十分震撼, 从理论上讲是可行的, 原理是利用地球外有一个配重。这个配重绕地球旋转的高度高于同步卫星轨道, 当它与地球同步转动时, 缆绳上保有张力使得电梯舱可以把物资运送到太空。关于相对地面静止在不同高度的物资，下列说法正确的是（ ） [图1] A: 物资在距离地心为地球半径处的线速度等于第一宇宙速度 B: 物资在配重空间站时处于完全失重状态 C: 物资所在高度越高, 受到电梯舱的弹力越小 D: 太空电梯上各点线速度与该点离地球球心的距离成正比

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: 《流浪地球 2 》中的太空电梯令人十分震撼, 从理论上讲是可行的, 原理是利用地球外有一个配重。这个配重绕地球旋转的高度高于同步卫星轨道, 当它与地球同步转动时, 缆绳上保有张力使得电梯舱可以把物资运送到太空。关于相对地面静止在不同高度的物资，下列说法正确的是（ ） [图1] A: 物资在距离地心为地球半径处的线速度等于第一宇宙速度 B: 物资在配重空间站时处于完全失重状态 C: 物资所在高度越高, 受到电梯舱的弹力越小 D: 太空电梯上各点线速度与该点离地球球心的距离成正比 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_449

人类对来知事物的好奇和科学家们的不懈努力, 使人类对宇宙的认识越来越丰富。 物体沿着圆周的运动是一种常见的运动, 匀速圆周运动是当中最简单也是最基本的一种, 由于做匀速圆周运动的物体的速度方向时刻在变化, 因而匀速圆周运动仍旧是一种变速运动。具有加速度, 可按如下模型来研究做匀速圆周运动的物体的加速度; 设质点沿半径为 $r$ 、圆心为 $O$ 的圆周以恒定大小的速度 $v$ 运动, 某时刻质点位于位置 $A$,经极短时间 $\Delta t$ 后运动到位置 $B$, 如图所示, 试根据加速度的定义, 推导质点在位置 $A$时的加速度的大小 $a_{A}$; [图1] 图1 [图2] 图2 [图3] 图3

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 人类对来知事物的好奇和科学家们的不懈努力, 使人类对宇宙的认识越来越丰富。 物体沿着圆周的运动是一种常见的运动, 匀速圆周运动是当中最简单也是最基本的一种, 由于做匀速圆周运动的物体的速度方向时刻在变化, 因而匀速圆周运动仍旧是一种变速运动。具有加速度, 可按如下模型来研究做匀速圆周运动的物体的加速度; 设质点沿半径为 $r$ 、圆心为 $O$ 的圆周以恒定大小的速度 $v$ 运动, 某时刻质点位于位置 $A$,经极短时间 $\Delta t$ 后运动到位置 $B$, 如图所示, 试根据加速度的定义, 推导质点在位置 $A$时的加速度的大小 $a_{A}$; [图1] 图1 [图2] 图2 [图3] 图3 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

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multi-modal

Astronomy_183

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, 当金星恰好运行到地球和太阳之间，且三者排成一条直线的现象，天文学称为“金星凌日”。当太阳位于金星和地球之间，且三者排成一条直线的现象，天文学称为“金星合日”。已知金星与太阳间的距离约为地球和太阳间距的 0.72 倍。下列判定正确的有 ( ) [图1] 地球 A: 金星绕太阳的周期约为 0.6 年 B: “金星凌日”的周期小于 1 年 C: 从某次“金星凌日”到最近的“金星合日”的时间小于 1 年 D: 从某次“金星凌日”到最近的“金星合日”的时间大于 1 年

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, 当金星恰好运行到地球和太阳之间，且三者排成一条直线的现象，天文学称为“金星凌日”。当太阳位于金星和地球之间，且三者排成一条直线的现象，天文学称为“金星合日”。已知金星与太阳间的距离约为地球和太阳间距的 0.72 倍。下列判定正确的有 ( ) [图1] 地球 A: 金星绕太阳的周期约为 0.6 年 B: “金星凌日”的周期小于 1 年 C: 从某次“金星凌日”到最近的“金星合日”的时间小于 1 年 D: 从某次“金星凌日”到最近的“金星合日”的时间大于 1 年 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_102

建造一条能通向太空的电梯 (如图甲所示), 是人们长期的梦想。材料的力学强度是材料众多性能中被人们极为看重的一种性能, 目前已发现的高强度材料碳纳米管的抗拉强度是钢的 100 倍, 密度是其 $\frac{1}{6}$, 这使得人们有望在赤道上建造垂直于水平面的“太空电梯”。图乙中 $r$ 为航天员到地心的距离, $R$ 为地球半径, $a-r$ 图像中的图线 $A$ 表示地球引力对航天员产生的加速度大小与 $r$ 的关系, 图线 $B$ 表示航天员由于地球自转而产生的向心加速度大小与 $r$ 的关系, 关于相对地面静止在不同高度的航天员, 地面附近重力加速度 $g$ 取 $10 \mathrm{~m} / \mathrm{s}^{2}$, 地球自转角速度 $\omega=7.3 \times 10^{-5} \mathrm{rad} / \mathrm{s}$, 地球半径 $R=6.4 \times 10^{3} \mathrm{~km}$ 。下列说法正确的有（） [图1] 图甲 [图2] 图乙 A: 随着 $r$ 增大, 航天员受到电梯舱的弹力减小 B: 航天员在 $r=R$ 处的线速度等于第一宇宙速度 C: 图中 $r_{0}$ 为地球同步卫星的轨道半径 D: 电梯舱停在距地面高度为 $5.6 R$ 的站点时, 舱内质量 $60 \mathrm{~kg}$ 的航天员对水平地板的压力为零

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: 建造一条能通向太空的电梯 (如图甲所示), 是人们长期的梦想。材料的力学强度是材料众多性能中被人们极为看重的一种性能, 目前已发现的高强度材料碳纳米管的抗拉强度是钢的 100 倍, 密度是其 $\frac{1}{6}$, 这使得人们有望在赤道上建造垂直于水平面的“太空电梯”。图乙中 $r$ 为航天员到地心的距离, $R$ 为地球半径, $a-r$ 图像中的图线 $A$ 表示地球引力对航天员产生的加速度大小与 $r$ 的关系, 图线 $B$ 表示航天员由于地球自转而产生的向心加速度大小与 $r$ 的关系, 关于相对地面静止在不同高度的航天员, 地面附近重力加速度 $g$ 取 $10 \mathrm{~m} / \mathrm{s}^{2}$, 地球自转角速度 $\omega=7.3 \times 10^{-5} \mathrm{rad} / \mathrm{s}$, 地球半径 $R=6.4 \times 10^{3} \mathrm{~km}$ 。下列说法正确的有（） [图1] 图甲 [图2] 图乙 A: 随着 $r$ 增大, 航天员受到电梯舱的弹力减小 B: 航天员在 $r=R$ 处的线速度等于第一宇宙速度 C: 图中 $r_{0}$ 为地球同步卫星的轨道半径 D: 电梯舱停在距地面高度为 $5.6 R$ 的站点时, 舱内质量 $60 \mathrm{~kg}$ 的航天员对水平地板的压力为零 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_677

2020 年 10 月 12 日和 26 日, 我国在西昌卫星发射中心分别将“高分十三号”和“天启星座 06 ”两颗地球卫星成功送入预定轨道。“高分十三号”是一颗高轨道光学遥感卫星, “天启星座 06 ”是一颗低轨道卫星, 若两卫星均绕地球做匀速圆周运动, 则由以上信息可知 ( ) A: “高分十三号”绕地球运动的周期小于“天启星座 06 ”的周期 B: “高分十三号”绕地球运动的动能小于“天启星座 06 ”的动能 C: “高分十三号”绕地球运动的加球度小于“天启星座 06 ”的加速度 D: “高分十三号”绕地球运动的角速度大于“天启星座 06 ”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。 这是一个单选题（只有一个正确答案）。 问题: 2020 年 10 月 12 日和 26 日, 我国在西昌卫星发射中心分别将“高分十三号”和“天启星座 06 ”两颗地球卫星成功送入预定轨道。“高分十三号”是一颗高轨道光学遥感卫星, “天启星座 06 ”是一颗低轨道卫星, 若两卫星均绕地球做匀速圆周运动, 则由以上信息可知 ( ) A: “高分十三号”绕地球运动的周期小于“天启星座 06 ”的周期 B: “高分十三号”绕地球运动的动能小于“天启星座 06 ”的动能 C: “高分十三号”绕地球运动的加球度小于“天启星座 06 ”的加速度 D: “高分十三号”绕地球运动的角速度大于“天启星座 06 ”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应为以下选项之一：[A, B, C, D]

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Astronomy

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text-only

Astronomy_905

Main sequence stars fuse hydrogen atoms to form helium in their cores. About $90 \%$ of the stars in the Universe, including the Sun, are main sequence stars. These stars can range from about a tenth of the mass of the Sun to up to 200 times as massive. The main source of energy in main sequence stars is from nuclear fusion. The mass of one hydrogen nucleus is $m_{\mathrm{H}}=1.674 \times 10^{-27} \mathrm{~kg}$, and the mass of one helium nucleus is $m_{\mathrm{He}}=6.649 \times 10^{-27} \mathrm{~kg}$. [figure1] Figure 3: Left: The proton-proton chain reaction is one of two known sets of nuclear fusion reactions by which stars convert hydrogen to helium. It dominates in stars with masses less than or equal to that of the Sun's, and involves a net change of four hydrogen nuclei becoming one helium nucleus. Right: Only the core of a main sequence star will undergo nuclear fusion due to the higher temperature than the surrounding hydrogen shell. By considering the difference in mass between the net inputs and outputs in the protonproton chain, how much energy is produced in this reaction? Hint: you will need to use Einstein's most famous equation.

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: Main sequence stars fuse hydrogen atoms to form helium in their cores. About $90 \%$ of the stars in the Universe, including the Sun, are main sequence stars. These stars can range from about a tenth of the mass of the Sun to up to 200 times as massive. The main source of energy in main sequence stars is from nuclear fusion. The mass of one hydrogen nucleus is $m_{\mathrm{H}}=1.674 \times 10^{-27} \mathrm{~kg}$, and the mass of one helium nucleus is $m_{\mathrm{He}}=6.649 \times 10^{-27} \mathrm{~kg}$. [figure1] Figure 3: Left: The proton-proton chain reaction is one of two known sets of nuclear fusion reactions by which stars convert hydrogen to helium. It dominates in stars with masses less than or equal to that of the Sun's, and involves a net change of four hydrogen nuclei becoming one helium nucleus. Right: Only the core of a main sequence star will undergo nuclear fusion due to the higher temperature than the surrounding hydrogen shell. By considering the difference in mass between the net inputs and outputs in the protonproton chain, how much energy is produced in this reaction? Hint: you will need to use Einstein's most famous equation. 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 J, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

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multi-modal

Astronomy_246

如图, 地球与月球可以看作双星系统, 它们均绕连线上的 $C$ 点转动, 在该系统的转动平面内有两个拉格朗日点 $L_{2} 、 L_{4}$, 位于这两个点的卫星能在地球引力和月球引力的共同作用下绕 $C$ 点做匀速圆周运动, 并保持与地球月球相对位置不变, $L_{2}$ 点在地月连线的延长线上, $L_{4}$ 点与地球球心、月球球心的连线构成一个等边三角形。我国已发射的“鹊桥”中继卫星位于 $L_{2}$ 点附近，它为“嫦娥四号”成功登陆月球背面提供了稳定的通信支持。假设 $L_{4}$ 点有一颗监测卫星, “鹊桥”中继卫星视为在 $L_{2}$ 点。已知地球的质量为月球的 81 倍, 则 ( ) [图1] A: 地球球心和月球球心到 $C$ 点的距离之比为 $81: 1$ B: 地球和月球对监测卫星的引力之比为 9: 1 C: 监测卫星绕 $C$ 点运行的加速度比月球的大 D: 监测卫星绕 $C$ 点运行的周期比“鹊桥”中继卫星的大

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 如图, 地球与月球可以看作双星系统, 它们均绕连线上的 $C$ 点转动, 在该系统的转动平面内有两个拉格朗日点 $L_{2} 、 L_{4}$, 位于这两个点的卫星能在地球引力和月球引力的共同作用下绕 $C$ 点做匀速圆周运动, 并保持与地球月球相对位置不变, $L_{2}$ 点在地月连线的延长线上, $L_{4}$ 点与地球球心、月球球心的连线构成一个等边三角形。我国已发射的“鹊桥”中继卫星位于 $L_{2}$ 点附近，它为“嫦娥四号”成功登陆月球背面提供了稳定的通信支持。假设 $L_{4}$ 点有一颗监测卫星, “鹊桥”中继卫星视为在 $L_{2}$ 点。已知地球的质量为月球的 81 倍, 则 ( ) [图1] A: 地球球心和月球球心到 $C$ 点的距离之比为 $81: 1$ B: 地球和月球对监测卫星的引力之比为 9: 1 C: 监测卫星绕 $C$ 点运行的加速度比月球的大 D: 监测卫星绕 $C$ 点运行的周期比“鹊桥”中继卫星的大 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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multi-modal

Astronomy_482

2021 年 4 月 29 日, 在海南文昌发射场用“长征五号”B 遥二运载火箭成功将中国空间站天和核心舱准确送入预定轨道。天和核心舱全长 $16.6 \mathrm{~m}$, 直径 $4.2 \mathrm{~m}$, 空间约 $50 \mathrm{~m}^{3}$, 距地运行高度 $400 \mathrm{~km}$ 。若地球半径 $R$ 地球表面重力加速度 $g$ 及引力常量 $G$ 已知,根据以上数据, 不能估算出的物理量是（） [图1] A: 地球的平均密度与第一宇宙速度 B: 天和核心舱的运行周期与线速度 C: 天和核心舱的动能与所受万有引力 D: 天和核心舱的角速度与向心加速度

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 2021 年 4 月 29 日, 在海南文昌发射场用“长征五号”B 遥二运载火箭成功将中国空间站天和核心舱准确送入预定轨道。天和核心舱全长 $16.6 \mathrm{~m}$, 直径 $4.2 \mathrm{~m}$, 空间约 $50 \mathrm{~m}^{3}$, 距地运行高度 $400 \mathrm{~km}$ 。若地球半径 $R$ 地球表面重力加速度 $g$ 及引力常量 $G$ 已知,根据以上数据, 不能估算出的物理量是（） [图1] A: 地球的平均密度与第一宇宙速度 B: 天和核心舱的运行周期与线速度 C: 天和核心舱的动能与所受万有引力 D: 天和核心舱的角速度与向心加速度 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应为以下选项之一：[A, B, C, D]

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Astronomy

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multi-modal

Astronomy_287

我国航天技术水平在世界处于领先地位，对于人造卫星的发射，有人提出了利用“地球隧道”发射人造卫星的构想：沿地球的一条弦挖一通道，在通道的两个出口处分别将等质量的待发射卫星部件同时释放，部件将在通道中间位置“碰撞组装”成卫星并静止下来; 另在通道的出口处由静止释放一个大质量物体，大质量物体会在通道与待发射的卫星碰撞, 只要物体质量相比卫星质量足够大, 卫星获得足够速度就会从对向通道口射出。 (以下计算中, 已知地球的质量为 $M_{0}$, 地球半径为 $R_{0}$, 引力常量为 $G$, 可忽略通道 $A B$的内径大小和地球自转影响。) 如图乙所示，设想在地球上距地心 $h$ 处沿弦长方向挖了一条光滑通道 $A B$, 一个质 量为 $m$ 。的质点在离通道中心 $O^{\prime}$ 的距离为 $x$ 处, 求质点所受万有引力沿弦 $A B$ 方向的分力 $F_{x}$; 将该质点从 $A$ 点静止释放, 求质点到达通道中心 $O^{\prime}$ 处时的速度大小 $v_{0}$ 。 [图1] 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: 我国航天技术水平在世界处于领先地位，对于人造卫星的发射，有人提出了利用“地球隧道”发射人造卫星的构想：沿地球的一条弦挖一通道，在通道的两个出口处分别将等质量的待发射卫星部件同时释放，部件将在通道中间位置“碰撞组装”成卫星并静止下来; 另在通道的出口处由静止释放一个大质量物体，大质量物体会在通道与待发射的卫星碰撞, 只要物体质量相比卫星质量足够大, 卫星获得足够速度就会从对向通道口射出。 (以下计算中, 已知地球的质量为 $M_{0}$, 地球半径为 $R_{0}$, 引力常量为 $G$, 可忽略通道 $A B$的内径大小和地球自转影响。) 如图乙所示，设想在地球上距地心 $h$ 处沿弦长方向挖了一条光滑通道 $A B$, 一个质 量为 $m$ 。的质点在离通道中心 $O^{\prime}$ 的距离为 $x$ 处, 求质点所受万有引力沿弦 $A B$ 方向的分力 $F_{x}$; 将该质点从 $A$ 点静止释放, 求质点到达通道中心 $O^{\prime}$ 处时的速度大小 $v_{0}$ 。 [图1] 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

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multi-modal

Astronomy_372

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 $\mathrm{F}$ 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(1)为近地圆轨道, 其轨道半径为 $R_{1}$, (2) 为椭圆变轨轨道, (3)为天和核心舱所在轨道,其轨道半径为 $R_{2}, P 、 Q$ 分别为(2)轨道与(1)、(3)轨道的交会点, 已知神舟十五号的质量为 $m$, 地球表面重力加速度为 $g$ 。关于神舟十五号载人飞船与天和核心舱交会对接过程, 下列说法正确的是（） [图1] A: 神舟十五号在(1)轨道上经过 $P$ 点时的机械能等于在(2)轨道上经过 $P$ 点时的机械能 B: 神舟十五号在(1)轨道上经过 $P$ 点时的加速度小于在(2)轨道上经过 $P$ 点时的加速度 C: 神舟十五号在(2)轨道上从 $P$ 点运动到 $Q$ 点经历的时间为 $t=\frac{\pi}{2}\left(1+\frac{R_{2}}{R_{1}}\right) \sqrt{\frac{R_{1}+R_{2}}{2 g}}$ D: 神舟十五号从(1)轨道转移到(3)轨道过程中, 飞船自身动力对飞船做的功为 $$ \frac{m g R_{1}\left(R_{2}-R_{1}\right)}{2 R_{2}} $$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 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 $\mathrm{F}$ 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(1)为近地圆轨道, 其轨道半径为 $R_{1}$, (2) 为椭圆变轨轨道, (3)为天和核心舱所在轨道,其轨道半径为 $R_{2}, P 、 Q$ 分别为(2)轨道与(1)、(3)轨道的交会点, 已知神舟十五号的质量为 $m$, 地球表面重力加速度为 $g$ 。关于神舟十五号载人飞船与天和核心舱交会对接过程, 下列说法正确的是（） [图1] A: 神舟十五号在(1)轨道上经过 $P$ 点时的机械能等于在(2)轨道上经过 $P$ 点时的机械能 B: 神舟十五号在(1)轨道上经过 $P$ 点时的加速度小于在(2)轨道上经过 $P$ 点时的加速度 C: 神舟十五号在(2)轨道上从 $P$ 点运动到 $Q$ 点经历的时间为 $t=\frac{\pi}{2}\left(1+\frac{R_{2}}{R_{1}}\right) \sqrt{\frac{R_{1}+R_{2}}{2 g}}$ D: 神舟十五号从(1)轨道转移到(3)轨道过程中, 飞船自身动力对飞船做的功为 $$ \frac{m g R_{1}\left(R_{2}-R_{1}\right)}{2 R_{2}} $$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

multi-modal

Astronomy_1115

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

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: It is often said that the Sun rises in the East and sets in the West, however this is only true twice a year at the equinoxes. In the Northern hemisphere, the Sun will rise northwards of East on the June solstice, and southwards of East on the December solstice; this is directly tied in with the varying length of day too, since the Sun either has a greater or shorter distance to travel across the sky (see Figure 1). [figure1] Figure 1: Left: The path of the Sun across the sky during the equinoxes and solstices, as viewed by an observer in the Northern hemisphere at a latitude of $\sim 40^{\circ}$. Credit: Daniel V. Schroeder / Weber State University. Right: The same idea but viewed from Iceland at a latitude of $65^{\circ}$, where by being so close to the Artic circle the day length can get close to 24 hours in June and almost no daylight in December. Credit: Kristn Bjarnadttir / University of Iceland. During the equinox, the Sun travels along the projection of the Earth's equator. In this question, we will assume a circular orbit for the Earth, and all angles will be calculated in degrees. A simple model for the vertical angle between the Sun and the horizon (known at the altitude), $h$, as a function of the bearing on the horizon, $A$ (measured clockwise from North, also called the azimuth), the latitude of the observer, $\phi$ (positive in Northern hemisphere, negative in Southern hemisphere), and the vertical angle of the Sun relative to the celestial equator (known as the solar declination), $\delta$, is given as: $$ h=-\left(90^{\circ}-\phi\right) \cos (A)+\delta $$ The solar declination can be considered to vary sinusoidally over the year, going from a maximum of $\delta=+23.44^{\circ}$ at the June solstice (roughly $21^{\text {st }}$ June) to a minimum of $\delta=-23.44^{\circ}$ on the December solstice (roughly $21^{\text {st }}$ December). It can be shown using spherical trigonometry that the precise model connecting $\delta, h, \phi$ and $A$ is: $$ \sin (\delta)=\sin (h) \sin (\phi)+\cos (h) \cos (\phi) \cos (A) . $$ Using the precise model, the path of the Sun across the sky forms a shape that is not quite the cosine shape of the simple model, and is shown in Figure 2. [figure2] Figure 2: The altitude of the Sun as a function of bearing during the equinoxes and solstices, as viewed by an observer at a latitude of $+56^{\circ}$. Whilst it resembles the cosine shape of the simple model well at this latitude, there are small deviations. Credit: Wikipedia. By using further spherical trigonometry, we can derive a second helpful equation in the precise model: $$ \sin (h)=\sin (\phi) \sin (\delta)+\cos (\phi) \cos (\delta) \cos (H) $$ Here, $H$ is the solar hour angle, which measures the angle between the Sun and solar noon as measured along the projection of the Earth's equator on the sky. Conventionally, $H=0^{\circ}$ at solar noon, is negative before solar noon, and is positive afterwards. Since the sun's hour angle increases at an approximately constant rate due to the rotation of the Earth, we can convert this angle into a time using the conversion $360^{\circ}=24^{\mathrm{h}}$. problem: b. Considering just the bearing of sunrise, suggest (with qualitative justification only) which of the following situations the simple model will be the best approximation for the precise model: A) a pole at solstice; B) a pole at equinox; C) the equator at solstice; or D) the equator at equinox. 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_919

When observing from the UK, during which season is the Full Moon visible highest in the sky? A: Spring B: Summer C: Autumn D: Winter

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: When observing from the UK, during which season is the Full Moon visible highest in the sky? A: Spring B: Summer C: Autumn D: Winter You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

SC

Astronomy

EN

text-only

Astronomy_1024

Light source A from the previous question is returned to a distance of $r$ from the detector. How far away should it now be moved to appear 5 magnitudes fainter? A: $5 r$ B: $10 r$ C: $50 \mathrm{r}$ D: $100 r$

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: Light source A from the previous question is returned to a distance of $r$ from the detector. How far away should it now be moved to appear 5 magnitudes fainter? A: $5 r$ B: $10 r$ C: $50 \mathrm{r}$ D: $100 r$ 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].

SC

Astronomy

EN

text-only

Astronomy_866

Leo then realizes that, in order for a single set of hour markings to accurately describe the time over the course of an entire year, the stick may need to be tilted away from the vertical position. More specifically, consider straight lines drawn on the ground from the base of the stick; the shadow at a certain fixed time of day, on different days of the year, should always lie on the same line. Measured as an angle from the vertical, how much does the stick need to be tilted, and in which direction? A: $0^{\circ}$ (no tilt needed) B: $40^{\circ}$ towards the North C: $50^{\circ}$ towards the North D: $40^{\circ}$ towards the South E: $50^{\circ}$ towards the South

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: Leo then realizes that, in order for a single set of hour markings to accurately describe the time over the course of an entire year, the stick may need to be tilted away from the vertical position. More specifically, consider straight lines drawn on the ground from the base of the stick; the shadow at a certain fixed time of day, on different days of the year, should always lie on the same line. Measured as an angle from the vertical, how much does the stick need to be tilted, and in which direction? A: $0^{\circ}$ (no tilt needed) B: $40^{\circ}$ towards the North C: $50^{\circ}$ towards the North D: $40^{\circ}$ towards the South E: $50^{\circ}$ towards the South 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_961

Estimate the number of photons incident on a human pupil (of radius $2 \mathrm{~mm}$ ) from the Sun per second when it is at the zenith on a clear day. A: $\sim 10^{10}$ B: $\sim 10^{13}$ C: $\sim 10^{16}$ D: $\sim 10^{19}$

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: Estimate the number of photons incident on a human pupil (of radius $2 \mathrm{~mm}$ ) from the Sun per second when it is at the zenith on a clear day. A: $\sim 10^{10}$ B: $\sim 10^{13}$ C: $\sim 10^{16}$ D: $\sim 10^{19}$ 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_1202

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

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is an expression. Here is some context information for this question, which might assist you in solving it: The surface of the Sun has a temperature of $\sim 5700 \mathrm{~K}$ yet the solar corona (a very faint region of plasma normally only visible from Earth during a solar eclipse) is considerably hotter at around $10^{6} \mathrm{~K}$. The source of coronal heating is a mystery and so understanding how this might happen is one of several key science objectives of the Solar Orbiter spacecraft. It is equipped with an array of cameras and will take photos of the Sun from distances closer than ever before (other probes will go closer, but none of those have cameras). [figure1] Figure 7: Left: The Sun's corona (coloured green) as viewed in visible light (580-640 nm) taken with the METIS coronagraph instrument onboard Solar Orbiter. The coronagraph is a disc that blocks out the light of the Sun (whose size and position is indicated with the white circle in the middle) so that the faint corona can be seen. This was taken just after first perihelion and is already at a resolution only matched by ground-based telescopes during a solar eclipse - once it gets into the main phase of the mission when it is even closer then its photos will be unrivalled. Credit: METIS Team / ESA \& NASA Right: A high-resolution image from the Extreme Ultraviolet Imager (EUI), taken with the $\mathrm{HRI}_{\mathrm{EUV}}$ telescope just before first perihelion. The circle in the lower right corner indicates the size of Earth for scale. The arrow points to one of the ubiquitous features of the solar surface, called 'campfires', that were discovered by this spacecraft and may play an important role in heating the corona. Credit: EUI Team / ESA \& NASA. Launched in February 2020 (and taken to be at aphelion at launch), it arrived at its first perihelion on $15^{\text {th }}$ June 2020 and has sent back some of the highest resolution images of the surface of the Sun (i.e. the base of the corona) we have ever seen. In them we have identified phenomena nicknamed as 'campfires' (see Fig 7) which are already being considered as a potential major contributor to the mechanism of coronal heating. Later on in its mission it will go in even closer, and so will take photos of the Sun in unprecedented detail. The highest resolution photos are taken with the Extreme Ultraviolet Imager (EUI), which consists of three separate cameras. One of them, the Extreme Ultraviolet High Resolution Imager (HRI $\mathrm{HUV}$ ), is designed to pick up an emission line from highly ionised atoms of iron in the corona. The iron being detected has lost 9 electrons (i.e. $\mathrm{Fe}^{9+}$ ) though is called $\mathrm{Fe} \mathrm{X} \mathrm{('ten')} \mathrm{by} \mathrm{astronomers} \mathrm{(as} \mathrm{Fe} \mathrm{I} \mathrm{is} \mathrm{the} \mathrm{neutral}$ atom). Its presence can be used to work out the temperature of the part of the corona being investigated by the instrument. The photons detected by $\mathrm{HRI}_{\mathrm{EUV}}$ are emitted by a rearrangement of the electrons in the $\mathrm{Fe} \mathrm{X}$ ion, corresponding to a photon energy of $71.0372 \mathrm{eV}$ (where $1 \mathrm{eV}=1.60 \times 10^{-19} \mathrm{~J}$ ). The HRI $\mathrm{HUV}_{\mathrm{EUV}}$ telescope has a $1000^{\prime \prime}$ by $1000^{\prime \prime}$ field of view (FOV, where $1^{\circ}=3600^{\prime \prime}=3600$ arcseconds), an entrance pupil diameter of $47.4 \mathrm{~mm}$, a couple of mirrors that give an effective focal length of $4187 \mathrm{~mm}$, and the image is captured by a CCD with 2048 by 2048 pixels, each of which is 10 by $10 \mu \mathrm{m}$. Although we are viewing the emissions of $\mathrm{Fe} \mathrm{X}$ ions, the vast majority of the plasma in the corona is hydrogen and helium, and the bulk motions of this determine the timescales over which visible phenomena change. In particular, the speed of sound is very important if we do not want motion blur to affect our high resolution images, as this sets the limit on exposure times. problem: c. The Rayleigh criterion and speed of sound in a plasma are given. ii. In practice, this is not achieved as the pixels are not small enough. Given that each picture element is spread across two pixels (in 1D) to allow adequate sampling, what is the actual minimum angle resolved on the CCD? Give your answer in arcseconds ("). [Hint: consider the geometry of the optical system and note that the angles are small enough that the small angle approximation can be used.] 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_950

When an exoplanet transits a star, the brightness of the star dims by 0.0557 magnitudes. Let $R_{P}$ be the radius of the planet and $R_{S}$ be the radius of the star. What is the ratio $R_{P} / R_{S}$ ? A: 0.00250 B: 0.224 C: 0.776 D: 0.998

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: When an exoplanet transits a star, the brightness of the star dims by 0.0557 magnitudes. Let $R_{P}$ be the radius of the planet and $R_{S}$ be the radius of the star. What is the ratio $R_{P} / R_{S}$ ? A: 0.00250 B: 0.224 C: 0.776 D: 0.998 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_1087

In science fiction films the asteroid belt is typically portrayed as a region of the Solar System where the spacecraft needs to dodge and weave its way through many large asteroids that are rather close together. However, if this image were true then very few probes would be able to pass through the belt into the outer Solar System. [figure1] Figure 1 Artist conceptual illustration of the asteroid belt (left). Schematic of the Solar System with the asteroid belt between Mars and Jupiter (right). This question will look at the real distances between asteroids.a. Given that the total mass of the asteroid belt is approximately $M_{\text {belt }}=1.8 \times 10^{-9} M_{\odot}$ calculate the radius of the object that could be formed, assuming it has a density typical of rock ( $\rho=3.0 \mathrm{~g} \mathrm{~cm}^{-3}$ ). Compare this to the radius of the largest member of the asteroid belt, Ceres. $\left(R_{\text {Ceres }}=473 \mathrm{~km}\right)$

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is an expression. Here is some context information for this question, which might assist you in solving it: In science fiction films the asteroid belt is typically portrayed as a region of the Solar System where the spacecraft needs to dodge and weave its way through many large asteroids that are rather close together. However, if this image were true then very few probes would be able to pass through the belt into the outer Solar System. [figure1] Figure 1 Artist conceptual illustration of the asteroid belt (left). Schematic of the Solar System with the asteroid belt between Mars and Jupiter (right). This question will look at the real distances between asteroids. problem: a. Given that the total mass of the asteroid belt is approximately $M_{\text {belt }}=1.8 \times 10^{-9} M_{\odot}$ calculate the radius of the object that could be formed, assuming it has a density typical of rock ( $\rho=3.0 \mathrm{~g} \mathrm{~cm}^{-3}$ ). Compare this to the radius of the largest member of the asteroid belt, Ceres. $\left(R_{\text {Ceres }}=473 \mathrm{~km}\right)$ All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is an expression without equals signs, e.g. ANSWER=\frac{1}{2} g t^2

EX

Astronomy

EN

multi-modal

Astronomy_398

如图所示, 已知“神舟十一号”从捕获“天宫二号”到实现对接用时为 $t$, 这段时间内组合体绕地球转过的角度为 $\theta$ (此过程轨道不变, 速度大小不变)。地球半径为 $R$, 地球表面重力加速度为 $g$, 引力常量为 $G$, 不考虑地球自转, 求组合体运动的周期 $T$ 及所在圆轨道离地高度 $H$ 。 [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, 已知“神舟十一号”从捕获“天宫二号”到实现对接用时为 $t$, 这段时间内组合体绕地球转过的角度为 $\theta$ (此过程轨道不变, 速度大小不变)。地球半径为 $R$, 地球表面重力加速度为 $g$, 引力常量为 $G$, 不考虑地球自转, 求组合体运动的周期 $T$ 及所在圆轨道离地高度 $H$ 。 [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_280

2021 年 2 月 10 日, 天问一号探测器成功实现近火制动开始绕火星运行, 2 月 15 日,天问一号探测器实现了完美的“侧手翻”, 将轨道调整为经过火星两极的环火星轨道。天问一号在绕火星运动过程中由于火星遮挡太阳光, 也会出现类似于地球上观察到的日全食现象, 如图所示。已知天问一号绕火星运动的轨道半径为 $r$, 火星质量为 $\mathrm{M}$, 引力常量为 $G$, 天问一号相对于火星的张角为 $\alpha$ (用弧度制表示), 将天问一号环火星看作匀速圆周运动, 天问一号、火星和太阳的球心在同一平面内, 太阳光可看作平行光, 则 [图1] A: 火星表面的重力加速度为 $\frac{G M}{r^{2} \sin ^{2} \alpha} \quad$ B: 火星的第一宇宙速度为 $\sqrt{\frac{G M}{r \tan \frac{\alpha}{2}}}$ C: 天问一号每次日全食持续的时间为 $\alpha \sqrt{\frac{r^{3}}{G M}}$ D: 天问一号运行的角速度为 $$ \sqrt{\frac{G M}{r^{3} \sin ^{3} \frac{\alpha}{2}}} $$

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: 2021 年 2 月 10 日, 天问一号探测器成功实现近火制动开始绕火星运行, 2 月 15 日,天问一号探测器实现了完美的“侧手翻”, 将轨道调整为经过火星两极的环火星轨道。天问一号在绕火星运动过程中由于火星遮挡太阳光, 也会出现类似于地球上观察到的日全食现象, 如图所示。已知天问一号绕火星运动的轨道半径为 $r$, 火星质量为 $\mathrm{M}$, 引力常量为 $G$, 天问一号相对于火星的张角为 $\alpha$ (用弧度制表示), 将天问一号环火星看作匀速圆周运动, 天问一号、火星和太阳的球心在同一平面内, 太阳光可看作平行光, 则 [图1] A: 火星表面的重力加速度为 $\frac{G M}{r^{2} \sin ^{2} \alpha} \quad$ B: 火星的第一宇宙速度为 $\sqrt{\frac{G M}{r \tan \frac{\alpha}{2}}}$ C: 天问一号每次日全食持续的时间为 $\alpha \sqrt{\frac{r^{3}}{G M}}$ D: 天问一号运行的角速度为 $$ \sqrt{\frac{G M}{r^{3} \sin ^{3} \frac{\alpha}{2}}} $$ 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

ZH

multi-modal

Astronomy_958

A consequence of the expansion of the Universe is that galaxies appear to be moving away from us, and the further they are away the quicker they seem to be moving. Edwin Hubble showed that if the expansion was uniform in all directions then the relationship can be expressed as $v=H_{0} d$ where $v$ is the recessional velocity, $d$ is the distance to the galaxy, and $H_{0}$ is called the Hubble constant. His original compilation of this data from his 1929 paper is shown in Figure 2. [figure1] Figure 2: The original plot of recessional velocity (based upon redshifts) given in $\mathrm{km} \mathrm{s}^{-1}$ (despite the incorrect axes labels) and distances given in $\mathrm{Mpc}\left(=10^{6} \mathrm{pc}\right.$ ) to 32 nearby galaxies. The solid line shows the best fit to the individual data points. Credit: Hubble (1929). Whilst the measured redshifts used to derive the values of $v$ are largely consistent with the modern values, the distances are considerably different from the ones we accept today. Even so, we can repeat the analysis he did to get a rough value for the age of the Universe, despite it being rather different from our current estimates. Making full use of the graph, calculate the value of $H_{0}$, giving your answer in units of $\mathrm{km} \mathrm{s}^{-1} \mathrm{Mpc}^{-1}$.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is an expression. problem: A consequence of the expansion of the Universe is that galaxies appear to be moving away from us, and the further they are away the quicker they seem to be moving. Edwin Hubble showed that if the expansion was uniform in all directions then the relationship can be expressed as $v=H_{0} d$ where $v$ is the recessional velocity, $d$ is the distance to the galaxy, and $H_{0}$ is called the Hubble constant. His original compilation of this data from his 1929 paper is shown in Figure 2. [figure1] Figure 2: The original plot of recessional velocity (based upon redshifts) given in $\mathrm{km} \mathrm{s}^{-1}$ (despite the incorrect axes labels) and distances given in $\mathrm{Mpc}\left(=10^{6} \mathrm{pc}\right.$ ) to 32 nearby galaxies. The solid line shows the best fit to the individual data points. Credit: Hubble (1929). Whilst the measured redshifts used to derive the values of $v$ are largely consistent with the modern values, the distances are considerably different from the ones we accept today. Even so, we can repeat the analysis he did to get a rough value for the age of the Universe, despite it being rather different from our current estimates. Making full use of the graph, calculate the value of $H_{0}$, giving your answer in units of $\mathrm{km} \mathrm{s}^{-1} \mathrm{Mpc}^{-1}$. 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/s/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 an expression without any units and equals signs, e.g. ANSWER=\frac{1}{2} g t^2

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Astronomy

EN

multi-modal

Astronomy_278

在某星球表面, 宇航员将一物块从距离地面高 $h$ 处, 从静止开始无初速度释放, 经时间 $t$ 落地。该星球可看作质量分布均匀的球体, 半径为 $R(h<<R)$, 万有引力常量为 $\mathrm{G}$, 不考虑星球的自转, 不计一切阻力, 则该星球的密度为 ( ) A: $\frac{3 h}{2 \pi G R t^{2}}$ B: $\frac{3 h}{4 \pi G R t^{2}}$ C: $\frac{3 h}{8 \pi G R t^{2}}$ D: $\frac{3 h}{16 \pi G R t^{2}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 在某星球表面, 宇航员将一物块从距离地面高 $h$ 处, 从静止开始无初速度释放, 经时间 $t$ 落地。该星球可看作质量分布均匀的球体, 半径为 $R(h<<R)$, 万有引力常量为 $\mathrm{G}$, 不考虑星球的自转, 不计一切阻力, 则该星球的密度为 ( ) A: $\frac{3 h}{2 \pi G R t^{2}}$ B: $\frac{3 h}{4 \pi G R t^{2}}$ C: $\frac{3 h}{8 \pi G R t^{2}}$ D: $\frac{3 h}{16 \pi G R t^{2}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

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text-only

Astronomy_26

发射地球同步卫星时, 先将卫星发射至近地圆轨道 1, 然后经点火, 使其沿椭圆轨道 2 运行, 最后再次点火, 将卫星送入同步圆轨道 3 , 轨道 1 和 2 相切于 $Q$ 点, 轨道 2 和 3 相切于 $P$ 点, 设卫星在 1 轨道和 3 轨道正常运行的速度和加速度分别为 $v_{1} 、 v_{3}$ 和 $a_{1} 、 a_{3}$, 在 2 轨道经过 $P$ 点时的速度和加速度为 $v_{2}$ 和 $a_{2}$, 且当卫星分别在 $1 、 2 、 3$ 轨道上正常运行时周期分别为 $T_{1} 、 T_{2} 、 T_{3}$, 以下说法正确的是 ( ) [图1] A: $v_{1}>v_{3}>v_{2}$ B: $v_{1}>v_{2}>v_{3}$ C: $a_{1}>a_{2}=a_{3}$ D: $T_{1}<T_{2}<T_{3}$

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: 发射地球同步卫星时, 先将卫星发射至近地圆轨道 1, 然后经点火, 使其沿椭圆轨道 2 运行, 最后再次点火, 将卫星送入同步圆轨道 3 , 轨道 1 和 2 相切于 $Q$ 点, 轨道 2 和 3 相切于 $P$ 点, 设卫星在 1 轨道和 3 轨道正常运行的速度和加速度分别为 $v_{1} 、 v_{3}$ 和 $a_{1} 、 a_{3}$, 在 2 轨道经过 $P$ 点时的速度和加速度为 $v_{2}$ 和 $a_{2}$, 且当卫星分别在 $1 、 2 、 3$ 轨道上正常运行时周期分别为 $T_{1} 、 T_{2} 、 T_{3}$, 以下说法正确的是 ( ) [图1] A: $v_{1}>v_{3}>v_{2}$ B: $v_{1}>v_{2}>v_{3}$ C: $a_{1}>a_{2}=a_{3}$ D: $T_{1}<T_{2}<T_{3}$ 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_720

发射地球同步卫星时, 先将卫星发射至近地圆轨道 1, 然后经点火, 使其沿粗圆轨道 2 运行, 最后再次点火, 将卫星送入同步圆轨道 3. 轨道 $1 、 2$ 相切于 $Q$ 点. 轨道 2、 3 相切于 $P$ 点 (如图), 则当卫星分别在 $1,2,3$, 轨道上正常运行时, 以下说法正确的是 $(\quad)$ [图1] A: 卫星在轨道 3 上的角速度小于在轨道 1 上的角速度 B: 卫星在轨道 2 上的周期小于在轨道 3 上的周期 C: 卫星在轨道 2 上经过 $P$ 点的速度小于在轨道 3 上经过 $P$ 点的速度 D: 卫星在轨道 1 上经过 $Q$ 点时的加速度大于它在轨道 2 上经过 $Q$ 点时的加速度

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: 发射地球同步卫星时, 先将卫星发射至近地圆轨道 1, 然后经点火, 使其沿粗圆轨道 2 运行, 最后再次点火, 将卫星送入同步圆轨道 3. 轨道 $1 、 2$ 相切于 $Q$ 点. 轨道 2、 3 相切于 $P$ 点 (如图), 则当卫星分别在 $1,2,3$, 轨道上正常运行时, 以下说法正确的是 $(\quad)$ [图1] A: 卫星在轨道 3 上的角速度小于在轨道 1 上的角速度 B: 卫星在轨道 2 上的周期小于在轨道 3 上的周期 C: 卫星在轨道 2 上经过 $P$ 点的速度小于在轨道 3 上经过 $P$ 点的速度 D: 卫星在轨道 1 上经过 $Q$ 点时的加速度大于它在轨道 2 上经过 $Q$ 点时的加速度 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

ZH

multi-modal

Astronomy_218

北斗卫星导航系统是中国自主研发、独立运行的全球卫星导航系统, 北斗卫星导航系统由空间段、地面段和用户段三部分组成。空间段包括 5 颗静止轨道卫星和 30 颗非静止轨道卫星。假设一颗非静止轨道卫星 $\mathrm{a}$ 在轨道上绕行 $n$ 圈所用时间为 $t$ 。如图所示。已知地球的半径为 $R$, 地球表面处的重力加速度为 $g$, 万有引力常量为 $G$, 求: 地球的第一宇宙速度 $v_{l}$; [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: 北斗卫星导航系统是中国自主研发、独立运行的全球卫星导航系统, 北斗卫星导航系统由空间段、地面段和用户段三部分组成。空间段包括 5 颗静止轨道卫星和 30 颗非静止轨道卫星。假设一颗非静止轨道卫星 $\mathrm{a}$ 在轨道上绕行 $n$ 圈所用时间为 $t$ 。如图所示。已知地球的半径为 $R$, 地球表面处的重力加速度为 $g$, 万有引力常量为 $G$, 求: 地球的第一宇宙速度 $v_{l}$; [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

ZH

multi-modal

Astronomy_462

有两颗人造地球卫星, 它们的质量之比 $m_{1}: m_{2}=1: 2$, 轨道半径之比 $r_{1}: r_{2}=1: 3$, 则它们的 ( ) A: 向心力大小之比 $F_{1}: F_{2}=2: 9$ B: 运行速率之比 $v_{1}: v_{2}=\sqrt{3}: 1$ C: 向心加速度大小之比 $a_{1}: a_{2}=1: 9$ D: 运行的周期之比 $T_{1}: T_{2}=3 \sqrt{3}: 1$

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: 有两颗人造地球卫星, 它们的质量之比 $m_{1}: m_{2}=1: 2$, 轨道半径之比 $r_{1}: r_{2}=1: 3$, 则它们的 ( ) A: 向心力大小之比 $F_{1}: F_{2}=2: 9$ B: 运行速率之比 $v_{1}: v_{2}=\sqrt{3}: 1$ C: 向心加速度大小之比 $a_{1}: a_{2}=1: 9$ D: 运行的周期之比 $T_{1}: T_{2}=3 \sqrt{3}: 1$ 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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text-only

Astronomy_763

The Olbers' paradox raises the following question: A: Is the universe infinitely large? B: Is the universe infinitely old? C: Are we alone in the universe? D: Why is the night sky dark?

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 Olbers' paradox raises the following question: A: Is the universe infinitely large? B: Is the universe infinitely old? C: Are we alone in the universe? D: Why is the night sky dark? 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

EN

text-only

Astronomy_1026

In May 1919 the British astronomers Frank Watson Dyson and Arthur Stanley Eddington organised for two teams to photograph a total solar eclipse, one from the West African island of Principe and the other from the Brazilian town of Sobral. Which aspect of a new scientific theory were their observations seen as decisive evidence for? A: The bending of light close to an object as predicted by Einstein's general relativity B: The existence of quantised electron energy levels as predicted by Bohr's model of the atom C: The Sun is undergoing nuclear fusion in its core as predicted by Eddington D: The solar corona has a strong magnetic field as predicted by Maxwell's equations

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 May 1919 the British astronomers Frank Watson Dyson and Arthur Stanley Eddington organised for two teams to photograph a total solar eclipse, one from the West African island of Principe and the other from the Brazilian town of Sobral. Which aspect of a new scientific theory were their observations seen as decisive evidence for? A: The bending of light close to an object as predicted by Einstein's general relativity B: The existence of quantised electron energy levels as predicted by Bohr's model of the atom C: The Sun is undergoing nuclear fusion in its core as predicted by Eddington D: The solar corona has a strong magnetic field as predicted by Maxwell's equations 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

EN

text-only

Astronomy_668

两颗卫星在同一平面内的同方向绕地球做匀速圆周运动, 卫星 $\mathrm{A}$ 是同步卫星, 卫星 $\mathrm{B}$ 的轨道半径是卫星 $\mathrm{A}$ 的 4 倍, 则卫星 $\mathrm{A}$ 与卫星 $\mathrm{B}$ 距离两次相邻最近的时间间隔约为 $(\quad)$ A: 0.5 天 B: 1 天 C: 4 天 D: $\frac{8}{7}$ 天

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: 两颗卫星在同一平面内的同方向绕地球做匀速圆周运动, 卫星 $\mathrm{A}$ 是同步卫星, 卫星 $\mathrm{B}$ 的轨道半径是卫星 $\mathrm{A}$ 的 4 倍, 则卫星 $\mathrm{A}$ 与卫星 $\mathrm{B}$ 距离两次相邻最近的时间间隔约为 $(\quad)$ A: 0.5 天 B: 1 天 C: 4 天 D: $\frac{8}{7}$ 天 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_812

Erez is designing a Newtonian telescope! The equation of the primary mirror is $y=x^{2} / 36 \mathrm{~m}-1 \mathrm{~m}$, and the telescope tube intersects the mirror at $y=0$. What is the f-number (focal ratio) of the telescope? A: $\mathrm{f} / 0.75$ B: $f / 1.00$ C: $\mathrm{f} / 1.25$ D: $\mathrm{f} / 1.33$ E: f/1.75

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: Erez is designing a Newtonian telescope! The equation of the primary mirror is $y=x^{2} / 36 \mathrm{~m}-1 \mathrm{~m}$, and the telescope tube intersects the mirror at $y=0$. What is the f-number (focal ratio) of the telescope? A: $\mathrm{f} / 0.75$ B: $f / 1.00$ C: $\mathrm{f} / 1.25$ D: $\mathrm{f} / 1.33$ E: f/1.75 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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text-only

Astronomy_117

地球质量为 $M$, 绕太阳做匀速圆周运动, 半径为 $R$, 有一质量为 $m$ 的飞船, 由静止开始从 $P$ 点在恒力 $F$ 的作用下, (不计飞船受到的万有引力), 沿 $P D$ 方向做匀加速直线运动, 一年后在 $D$ 点飞船掠过地球上空, 再过三个月, 又在 $Q$ 处掠过地球上空. (取 $\pi^{2}=10$ )根据以上条件可以得出（） [图1] A: $D Q$ 的距离为 $\sqrt{2} R$ B: $P D$ 的距离为 $\frac{16 \sqrt{2}}{9} R$ C: 地球与太阳的万有引力的大小 $\frac{90 \sqrt{2} F M}{16 m}$ D: 地球与太阳的万有引力的大小 $\frac{9 \sqrt{3} F M}{16 m}$

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: 地球质量为 $M$, 绕太阳做匀速圆周运动, 半径为 $R$, 有一质量为 $m$ 的飞船, 由静止开始从 $P$ 点在恒力 $F$ 的作用下, (不计飞船受到的万有引力), 沿 $P D$ 方向做匀加速直线运动, 一年后在 $D$ 点飞船掠过地球上空, 再过三个月, 又在 $Q$ 处掠过地球上空. (取 $\pi^{2}=10$ )根据以上条件可以得出（） [图1] A: $D Q$ 的距离为 $\sqrt{2} R$ B: $P D$ 的距离为 $\frac{16 \sqrt{2}}{9} R$ C: 地球与太阳的万有引力的大小 $\frac{90 \sqrt{2} F M}{16 m}$ D: 地球与太阳的万有引力的大小 $\frac{9 \sqrt{3} F M}{16 m}$ 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_800

Which of the following is a problem of the conventional Big Bang theory that is resolved by the theory of inflation? A: Under the conventional Big Bang theory, it is extremely unlikely for our universe to be flat or nearly flat today, contrary to observation. B: Under the conventional Big Bang theory, it is impossible for the Cosmic Microwave Background to have come into thermal equilibrium by the time of recombination, despite its observed uniform temperature. C: The conventional Big Bang theory predicts a huge abundance of magnetic monopoles, while no magnetic monopoles have ever been discovered. D: All of the above

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 of the following is a problem of the conventional Big Bang theory that is resolved by the theory of inflation? A: Under the conventional Big Bang theory, it is extremely unlikely for our universe to be flat or nearly flat today, contrary to observation. B: Under the conventional Big Bang theory, it is impossible for the Cosmic Microwave Background to have come into thermal equilibrium by the time of recombination, despite its observed uniform temperature. C: The conventional Big Bang theory predicts a huge abundance of magnetic monopoles, while no magnetic monopoles have ever been discovered. D: All of the above 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

EN

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Astronomy_443

如图所示是北斗导航系统中部分卫星的轨道示意图，已知 $a 、 b 、 c$ 三颗卫星均做圆周运动, $a$ 是地球同步卫星, 则 ( ) [图1] A: 卫星 $a$ 的运行速度小于 $c$ 的运行速度 B: 卫星 $a$ 的加速度大于 $c$ 的加速度 C: 卫星 $b$ 的运行速度大于第一宇宙速度 D: 卫星 $c$ 的周期大于 $24 \mathrm{~h}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 如图所示是北斗导航系统中部分卫星的轨道示意图，已知 $a 、 b 、 c$ 三颗卫星均做圆周运动, $a$ 是地球同步卫星, 则 ( ) [图1] A: 卫星 $a$ 的运行速度小于 $c$ 的运行速度 B: 卫星 $a$ 的加速度大于 $c$ 的加速度 C: 卫星 $b$ 的运行速度大于第一宇宙速度 D: 卫星 $c$ 的周期大于 $24 \mathrm{~h}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

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