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Astronomy_627

银河系中由一颗白矮星和它的类日伴星组成的双星系统, 由于白矮星不停的吸收类日伴星抛出的物质致使其质量不断增加, 假设类日伴星所释放的物质被白矮星全部吸收,并且两星之间的距离在一段时间内不变, 两星球的总质量不变, 则下列说法正确的是 A: 两星间的万有引力不变 B: 两星的运动周期不变 C: 类日伴星的轨道半径增大 D: 白矮星的轨道半径增大

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 银河系中由一颗白矮星和它的类日伴星组成的双星系统, 由于白矮星不停的吸收类日伴星抛出的物质致使其质量不断增加, 假设类日伴星所释放的物质被白矮星全部吸收,并且两星之间的距离在一段时间内不变, 两星球的总质量不变, 则下列说法正确的是 A: 两星间的万有引力不变 B: 两星的运动周期不变 C: 类日伴星的轨道半径增大 D: 白矮星的轨道半径增大 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

text-only

Astronomy_226

随着科技的发展, 人类必将揭开火星的神秘面纱.如图所示, 火星的人造卫星在火星赤道的正上方距离火星表面高度为 $R$ 处环绕火星做匀速圆周运动, 已知卫星的运行方向与火星的自转方向相同, $a$ 点为火星赤道上的点, 该点有一接收器, 可接收到卫星发出信号。已知火星的半径为 $R$, 火星同步卫星的周期为 $T$, 近火卫星的线速度为 $v$, 引力常量为 $G$ 。则下列说法正确的是 ( ) [图1] A: 火星的质量为 $\frac{G}{v^{2} R}$ B: 卫星的环绕周期为 $\frac{4 \pi R}{v}$ C: $a$ 点连续收到信号的最长时间为 $\frac{4 \sqrt{2} \pi R}{3 v}$ D: 火星同步卫星到火星表面的高度为 $\sqrt[3]{\frac{v^{2} T^{2} R}{4 \pi^{2}}}-R$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 随着科技的发展, 人类必将揭开火星的神秘面纱.如图所示, 火星的人造卫星在火星赤道的正上方距离火星表面高度为 $R$ 处环绕火星做匀速圆周运动, 已知卫星的运行方向与火星的自转方向相同, $a$ 点为火星赤道上的点, 该点有一接收器, 可接收到卫星发出信号。已知火星的半径为 $R$, 火星同步卫星的周期为 $T$, 近火卫星的线速度为 $v$, 引力常量为 $G$ 。则下列说法正确的是 ( ) [图1] A: 火星的质量为 $\frac{G}{v^{2} R}$ B: 卫星的环绕周期为 $\frac{4 \pi R}{v}$ C: $a$ 点连续收到信号的最长时间为 $\frac{4 \sqrt{2} \pi R}{3 v}$ D: 火星同步卫星到火星表面的高度为 $\sqrt[3]{\frac{v^{2} T^{2} R}{4 \pi^{2}}}-R$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

multi-modal

Astronomy_1188

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.b. The energy density of black-body radiation, $u$, and number density, $n$, at temperature $T$ are given. ii. Assuming the plasma of Fe ions is in thermal equilibrium with the photons, and that the average energy of the photons is equal to the ionisation energy of FeX (which is $22540 \mathrm{~kJ}$ $\mathrm{mol}^{-1}$ ), calculate the temperature of the plasma. Give your answer to 4 s.f.

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 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: b. The energy density of black-body radiation, $u$, and number density, $n$, at temperature $T$ are given. ii. Assuming the plasma of Fe ions is in thermal equilibrium with the photons, and that the average energy of the photons is equal to the ionisation energy of FeX (which is $22540 \mathrm{~kJ}$ $\mathrm{mol}^{-1}$ ), calculate the temperature of the plasma. Give your answer to 4 s.f. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of \mathrm{~K}, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

multi-modal

Astronomy_402

我国的“天链一号”卫星是地球同步卫星, 可为中低轨道卫星提供数据通讯, 如图为 “天链一号”卫星 $a$ 、赤道平面内的低轨道卫星 $b$ 、地球的位置关系示意图, $O$ 为地心，地球相对卫星 $a 、 b$ 的张角分别为 $\theta_{1}$ 和 $\theta_{2}\left(\theta_{2}\right.$ 图中未标出), 卫星 $a$ 的轨道半径是 $b$ 的 4 倍, 已知卫星 $a 、 b$ 绕地球同向运行, 卫星 $a$ 的周期为 $T$, 在运行过程中由于地球的遮挡, 卫星 $b$ 会进入卫星 $a$ 通讯的盲区，卫星间的通讯信号视为沿直线传播，信号传输时间可忽略. 下列分析正确的是 ( ) [图1] A: 卫星 $a, b$ 的速度之比为 $2: 1$ B: 卫星 $b$ 的周期为 $T / 8$ C: 卫星 $b$ 每次在盲区运行的时间为 $\frac{\theta_{1}+\theta_{2}}{14 \pi} T$ D: 卫星 $b$ 每次在盲区运行的时间为 $\frac{\theta_{1}+\theta_{2}}{16 \pi} T$

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

MC

Astronomy

ZH

multi-modal

Astronomy_593

宇宙中有两颗孤立的中子星, 它们在相互的万有引力作用下间距保持不变, 并沿半径不同的同心圆轨道做匀速圆周运动. 如果双星间距为 $L$, 质量分别为 $m_{1}$ 和 $m_{2}$, 引力常量为 $G$, 则 ( ) A: 双星中 $m_{1}$ 的轨道半径 $r_{1}=\frac{m_{2}}{m_{1}+m_{2}} L$ B: 双星的运行周期 $T=2 \pi L \sqrt{\frac{m_{2} L}{G\left(m_{1}+m_{2}\right)}}$ C: $m_{1}$ 的线速度大小 $v_{1}=m_{1} \sqrt{\frac{G}{L\left(m_{1}+m_{2}\right)}}$ D: 若周期为 $T$, 则总质量 $m_{1}+m_{2}=\frac{4 \pi^{2} L^{3}}{G T^{2}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 宇宙中有两颗孤立的中子星, 它们在相互的万有引力作用下间距保持不变, 并沿半径不同的同心圆轨道做匀速圆周运动. 如果双星间距为 $L$, 质量分别为 $m_{1}$ 和 $m_{2}$, 引力常量为 $G$, 则 ( ) A: 双星中 $m_{1}$ 的轨道半径 $r_{1}=\frac{m_{2}}{m_{1}+m_{2}} L$ B: 双星的运行周期 $T=2 \pi L \sqrt{\frac{m_{2} L}{G\left(m_{1}+m_{2}\right)}}$ C: $m_{1}$ 的线速度大小 $v_{1}=m_{1} \sqrt{\frac{G}{L\left(m_{1}+m_{2}\right)}}$ D: 若周期为 $T$, 则总质量 $m_{1}+m_{2}=\frac{4 \pi^{2} L^{3}}{G T^{2}}$ 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应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

text-only

Astronomy_826

Here is a map of MIT and the surrounding area, where North points directly upwards, as taken from https://whereis.mit.edu: [figure1] Leo is biking along the Harvard Bridge (marked as "A") when he stops and looks out at the river. Looking out downriver (to the right on this map) and parallel to the banks, he sees the Sun straight in front of him, peeking out from above the buildings, and has to avert his eyes to not be blinded. What part of the academic year is it? A: Early fall semester (late September-early October) B: Late fall semester (late November-early December) C: Independent Activities Period (January) D: Early spring semester (late February-early March) E: Late spring semester (late April-early May)

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: Here is a map of MIT and the surrounding area, where North points directly upwards, as taken from https://whereis.mit.edu: [figure1] Leo is biking along the Harvard Bridge (marked as "A") when he stops and looks out at the river. Looking out downriver (to the right on this map) and parallel to the banks, he sees the Sun straight in front of him, peeking out from above the buildings, and has to avert his eyes to not be blinded. What part of the academic year is it? A: Early fall semester (late September-early October) B: Late fall semester (late November-early December) C: Independent Activities Period (January) D: Early spring semester (late February-early March) E: Late spring semester (late April-early May) You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D, E].

SC

Astronomy

EN

multi-modal

Astronomy_211

牛顿在 1689 年出版的《自然哲学的数学原理》中设想, 物体抛出的速度很大时, 就不会落回地面, 它将绕地球运动, 成为人造地球卫星。如图所示, 将物体从一座高山上 的 $O$ 点水平抛出, 抛出速度一次比一次大, 落地点一次比一次远, 设图中 $A 、 B 、 C 、$ $D 、 E$ 是从 $O$ 点以不同的速度抛出的物体所对应的运动轨道。已知 $B$ 是圆形轨道, $C 、 D$是椭圆轨道, 在轨道 $E$ 上运动的物体将会克服地球的引力, 永远地离开地球, 空气阻力和地球自转的影响不计, 则下列说法正确的是 ( ) [图1] A: 物体从 $O$ 点抛出后, 沿轨道 $A$ 运动落到地面上, 物体的运动可能是平抛运动 B: 在轨道 $B$ 上运动的物体, 抛出时的速度大小为 $7.9 \mathrm{~km} / \mathrm{s}$ C: 使轨道 $C 、 D$ 上物体的运动轨道变为圆轨道, 这个圆轨道可以过 $O$ 点 D: 在轨道 $E$ 上运动的物体，抛出时的速度一定等于或大于第三宇宙速度

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: 牛顿在 1689 年出版的《自然哲学的数学原理》中设想, 物体抛出的速度很大时, 就不会落回地面, 它将绕地球运动, 成为人造地球卫星。如图所示, 将物体从一座高山上 的 $O$ 点水平抛出, 抛出速度一次比一次大, 落地点一次比一次远, 设图中 $A 、 B 、 C 、$ $D 、 E$ 是从 $O$ 点以不同的速度抛出的物体所对应的运动轨道。已知 $B$ 是圆形轨道, $C 、 D$是椭圆轨道, 在轨道 $E$ 上运动的物体将会克服地球的引力, 永远地离开地球, 空气阻力和地球自转的影响不计, 则下列说法正确的是 ( ) [图1] A: 物体从 $O$ 点抛出后, 沿轨道 $A$ 运动落到地面上, 物体的运动可能是平抛运动 B: 在轨道 $B$ 上运动的物体, 抛出时的速度大小为 $7.9 \mathrm{~km} / \mathrm{s}$ C: 使轨道 $C 、 D$ 上物体的运动轨道变为圆轨道, 这个圆轨道可以过 $O$ 点 D: 在轨道 $E$ 上运动的物体，抛出时的速度一定等于或大于第三宇宙速度 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_276

人类太空探测计划旨在探测恒星亮度以寻找适合人类居住的宜居行星。在某次探测中发现距地球数光年处有一颗相对太阳静止的质量为 $M$ 的恒星 $\mathrm{A}$, 将恒星 $\mathrm{A}$ 视为黑体,根据斯特藩-玻尔兹曼定律: 一个黑体表面单位面积辐射出的功率与黑体本身的热力学温度 $T$ 的四次方成正比, 即黑体表面单位面积辐射出的功率为 $\sigma T^{4}$ (其中 $\sigma$ 为常数), A 的表面温度为 $T_{0}$, 地球上正对 $\mathrm{A}$ 的单位面积接收到 $\mathrm{A}$ 辐射出的功率为 $I$ 。已知 $\mathrm{A}$ 在地球轨道平面上，地球公转半径为 $R_{0}$, 一年内地球上的观测者测得地球与 $\mathrm{A}$ 的连线之间的最大夹角为 $\theta$ (角 $\theta$ 很小, 可认为 $\sin \theta \approx \tan \theta \approx \theta$ )。恒星 $\mathrm{A}$ 有一颗绕它做匀速圆周运动的行星 $\mathrm{B}$, 该行星也可视为黑体, 其表面的温度保持为 $T_{1}$, 恒星 $\mathrm{A}$ 射向行星 $\mathrm{B}$ 的光可看作平行光。已知引力常量为 $G$, 求: 行星 $\mathrm{B}$ 的运动周期 $T_{\mathrm{B}}$ 。

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 人类太空探测计划旨在探测恒星亮度以寻找适合人类居住的宜居行星。在某次探测中发现距地球数光年处有一颗相对太阳静止的质量为 $M$ 的恒星 $\mathrm{A}$, 将恒星 $\mathrm{A}$ 视为黑体,根据斯特藩-玻尔兹曼定律: 一个黑体表面单位面积辐射出的功率与黑体本身的热力学温度 $T$ 的四次方成正比, 即黑体表面单位面积辐射出的功率为 $\sigma T^{4}$ (其中 $\sigma$ 为常数), A 的表面温度为 $T_{0}$, 地球上正对 $\mathrm{A}$ 的单位面积接收到 $\mathrm{A}$ 辐射出的功率为 $I$ 。已知 $\mathrm{A}$ 在地球轨道平面上，地球公转半径为 $R_{0}$, 一年内地球上的观测者测得地球与 $\mathrm{A}$ 的连线之间的最大夹角为 $\theta$ (角 $\theta$ 很小, 可认为 $\sin \theta \approx \tan \theta \approx \theta$ )。恒星 $\mathrm{A}$ 有一颗绕它做匀速圆周运动的行星 $\mathrm{B}$, 该行星也可视为黑体, 其表面的温度保持为 $T_{1}$, 恒星 $\mathrm{A}$ 射向行星 $\mathrm{B}$ 的光可看作平行光。已知引力常量为 $G$, 求: 行星 $\mathrm{B}$ 的运动周期 $T_{\mathrm{B}}$ 。 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是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

text-only

Astronomy_513

一颗侦察卫星在通过地球两极上空的圆轨道上运行, 它的运行轨道距地面高度为 $h$ 。设地球半径为 $R$, 地面重力加速度为 $g$, 地球自转的周期为 $T$ 。要使该卫星在一天的时间内将地面上赤道各处在日照条件下的全部情况全都拍摄下来, 则卫星在通过赤道上空时, 卫星上的摄像机应拍摄地面上赤道圆周的弧长至少为（ ） A: $\frac{4 \pi^{2}(R+h)}{T} \sqrt{\frac{R+h}{g}}$ B: $\frac{2 \pi(R+h)}{T} \sqrt{\frac{R+h}{g}}$ C: $\frac{2 \pi R}{T} \sqrt{g}$ D: $\frac{4 \pi^{2}(R+h)}{T} \sqrt{g}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 一颗侦察卫星在通过地球两极上空的圆轨道上运行, 它的运行轨道距地面高度为 $h$ 。设地球半径为 $R$, 地面重力加速度为 $g$, 地球自转的周期为 $T$ 。要使该卫星在一天的时间内将地面上赤道各处在日照条件下的全部情况全都拍摄下来, 则卫星在通过赤道上空时, 卫星上的摄像机应拍摄地面上赤道圆周的弧长至少为（ ） A: $\frac{4 \pi^{2}(R+h)}{T} \sqrt{\frac{R+h}{g}}$ B: $\frac{2 \pi(R+h)}{T} \sqrt{\frac{R+h}{g}}$ C: $\frac{2 \pi R}{T} \sqrt{g}$ D: $\frac{4 \pi^{2}(R+h)}{T} \sqrt{g}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

ZH

text-only

Astronomy_304

被誉为“中国天眼”的射电望远镜 $F A \mathrm{~S} T$ 自工作以来, 已经发现 43 颗脉冲星, 为我国天文观测做出了巨大的贡献。脉冲星实质是快速自转的中子星, 中子星每自转一周,它的磁场就会扫过地球一次，地球就会接收到一个射电脉冲。若观测到某个中子星的射电脉冲周期为 $T$, 中子星两极处的重力加速度为 $g$, 密度为 $\rho$, 引力常量为 $G$ 。下列说法正确的是（） A: 中子星的半径为 $\frac{g T^{2}}{4 \pi^{2}}$ B: 中子星的质量为 $\frac{9 g^{3}}{16 \pi^{2} G^{2} \rho^{2}}$ C: 中子星的第一宇宙速度为 $\sqrt{\frac{3 g}{4 \pi G \rho}}$ D: 若地球接收射电脉冲的周期变为 $\sqrt{\frac{3 \pi}{G \rho}}$, 则中子星赤道上的物体会离开中子星表面

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 被誉为“中国天眼”的射电望远镜 $F A \mathrm{~S} T$ 自工作以来, 已经发现 43 颗脉冲星, 为我国天文观测做出了巨大的贡献。脉冲星实质是快速自转的中子星, 中子星每自转一周,它的磁场就会扫过地球一次，地球就会接收到一个射电脉冲。若观测到某个中子星的射电脉冲周期为 $T$, 中子星两极处的重力加速度为 $g$, 密度为 $\rho$, 引力常量为 $G$ 。下列说法正确的是（） A: 中子星的半径为 $\frac{g T^{2}}{4 \pi^{2}}$ B: 中子星的质量为 $\frac{9 g^{3}}{16 \pi^{2} G^{2} \rho^{2}}$ C: 中子星的第一宇宙速度为 $\sqrt{\frac{3 g}{4 \pi G \rho}}$ D: 若地球接收射电脉冲的周期变为 $\sqrt{\frac{3 \pi}{G \rho}}$, 则中子星赤道上的物体会离开中子星表面 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

ZH

text-only

Astronomy_953

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

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

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Astronomy

EN

multi-modal

Astronomy_1080

In order to move a spacecraft between orbits we must apply a thrust using rockets, which changes the velocity of the spacecraft by $\Delta v$. In this question we will ignore changes in the mass of the spacecraft due to the burning of fuel. For an object of mass $m$ in a circular orbit of radius $r$ around an object with mass $M$ (where $m \ll M$ ) the orbital velocity, $v_{\text {orb }}$, is given by the formula $v_{\text {orb }}=\sqrt{\frac{G M}{r}}$.f. How long would any astronauts on board the spacecraft need to wait until they could use a Hohmann transfer orbit to return to Earth? Hence calculate the total duration of the mission.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: In order to move a spacecraft between orbits we must apply a thrust using rockets, which changes the velocity of the spacecraft by $\Delta v$. In this question we will ignore changes in the mass of the spacecraft due to the burning of fuel. For an object of mass $m$ in a circular orbit of radius $r$ around an object with mass $M$ (where $m \ll M$ ) the orbital velocity, $v_{\text {orb }}$, is given by the formula $v_{\text {orb }}=\sqrt{\frac{G M}{r}}$. problem: f. How long would any astronauts on board the spacecraft need to wait until they could use a Hohmann transfer orbit to return to Earth? Hence calculate the total duration of the mission. 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 days, 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

EN

text-only

Astronomy_1134

The James Webb Space Telescope (JWST) is an incredibly exciting next generation telescope that was successfully launched on $25^{\text {th }}$ December 2021 . Its mirror is approximately $6.5 \mathrm{~m}$ in diameter, much larger than the $2.4 \mathrm{~m}$ mirror of the Hubble Space Telescope (HST), and so it has far greater resolution and sensitivity. Whilst HST largely imaged in the visible, JWST will do most of its work in the nearand mid-infrared (NIR and MIR respectively). This will allow it to pick up heavily redshifted light, such as that from the first generation of stars in the very first galaxies. [figure1] Figure 5: Left: A full-scale model of JWST next to some of the scientists and engineers involved in its development at the Goddard Space Flight Center. Credit: NASA / Goddard Space Flight Center / Pat Izzo. Right: The position of the second Lagrangian point, $L_{2}$, relative to the Earth. Credit: ESA. The resolution limit of a telescope is set by the amount of diffraction light rays experience as they enter the system, and is related to the diameter of a telescope, $D$, and the wavelength being observed, $\lambda$. The resolution limit of a CCD is set by the size of the pixels. Three of the imaging cameras on JWST are tabulated with some properties below: | Instrument | Wavelength range $(\mu \mathrm{m})$ | CCD plate scale (arcseconds / pixel) | | :---: | :---: | :---: | | NIRCam (short wave) | $0.6-2.3$ | 0.031 | | NIRCam (long wave) | $2.4-5.0$ | 0.065 | | MIRI | $5.6-25.5$ | 0.11 | An arcsecond is a measure of angle where $1^{\circ}=3600$ arcseconds. The familiar variation in intensity on a screen, $I_{\text {slit }}$, due to diffraction through an infinitely tall single slit is given as $$ I_{\text {slit }}=I_{0}\left(\frac{\sin (x)}{x}\right)^{2}, \text { where } \quad x=\frac{\pi D \theta}{\lambda} $$ and $I_{0}$ is the initial intensity. For a circular aperture, the formula is slightly different and is given as $$ I_{\mathrm{circ}}=I_{0}\left(\frac{2 J_{1}(x)}{x}\right)^{2} . $$ Here $J_{1}(x)$ is the Bessel function of the first kind and is calculated as $$ J_{n}(x)=\sum_{r=0}^{\infty} \frac{(-1)^{r}}{r !(n+r) !}\left(\frac{x}{2}\right)^{n+2 r} \quad \text { so } \quad J_{1}(x)=\frac{x}{2}\left(1-\frac{x^{2}}{8}+\frac{x^{4}}{192}-\ldots\right) . $$ The $x$-axis intercepts and shape of the maxima are quite different, as shown in Figure 6. The position of the first minimum of $I_{\text {slit }}$ is at $x_{\min }=\pi$ meaning that $\theta_{\min , \text { slit }}=\lambda / D$, whilst for $I_{\text {circ }}$ it is at $x_{\min }=3.8317 \ldots$ so $\theta_{\min , \mathrm{circ}} \approx 1.22 \lambda / D$. This is one way of defining the minimum angular resolution, although since the flux drops off so steeply away from the central maximum a more convenient one for use with CCDs is the angle corresponding to the full width half maximum (FWHM). [figure2] Figure 6: Left: The $I_{\text {slit }}$ (purple) and $I_{\text {circ }}$ (blue - the wider central maximum) functions, normalised so that $I_{0}=1$. You can see the shapes and $x$-intercepts are different. Right: How $x_{\min }$ and the full width half maximum (FWHM) are defined. Here it is shown for $I_{\text {circ }}$. As well as having the largest mirror of any space telescope ever launched, it is also one of the most sensitive, with its greatest sensitivity in the NIRCam F200W filter (centred on a wavelength of $1.989 \mu \mathrm{m})$ where after $10^{4}$ seconds it can detect a flux of $9.1 \mathrm{nJy}\left(1 \mathrm{Jy}=10^{-26} \mathrm{~W} \mathrm{~m}^{-2} \mathrm{~Hz}^{-1}\right.$ ) with a signal-to-noise ratio (S/N) of 10 , corresponding to an apparent magnitude of $m=29.0$. This extraordinary sensitivity can be used to pick up light from the earliest galaxies in the Universe. The scale factor, $a$, parameterises the expansion of the Universe since the Big Bang, and is related to the redshift, $z$, as $$ a=(1+z)^{-1} \quad \text { where } \quad z \equiv \frac{\lambda_{\text {obs }}-\lambda_{\mathrm{emit}}}{\lambda_{\mathrm{emit}}} $$ with $\lambda_{\text {obs }}$ the observed wavelength and $\lambda_{\text {emit }}$ the rest frame wavelength. The current rate of expansion of the Universe is given by the Hubble constant, $H_{0}$, and this is related to the current Hubble time, $t_{\mathrm{H}_{0}}$, and current Hubble distance, $D_{\mathrm{H}_{0}}$, as $$ t_{\mathrm{H}_{0}} \equiv H_{0}^{-1} \quad \text { and } \quad D_{\mathrm{H}_{0}} \equiv c t_{\mathrm{H}_{0}} \text {. } $$ Here the subscript 0 indicates the values are measured today. The Hubble constant is more appropriately known as the Hubble parameter as it is a function of time, and the evolution of $H$ as a function of $z$ is $$ E(z)=\frac{H}{H_{0}} \equiv\left[\Omega_{0, m}(1+z)^{3}+\Omega_{0, \Lambda}+\Omega_{0, r}(1+z)^{4}\right]^{1 / 2}, $$ where $\Omega$ is the normalised density parameter, and the subscript $m, r$, and $\Lambda$ indicate the contribution to $\Omega$ from matter, radiation, and dark energy, respectively. The proper age of the Universe $t(z)$ at redshift $z$ is best evaluated in terms of $a$ as $$ t=t_{\mathrm{H}_{0}} \int_{0}^{(1+z)^{-1}} \frac{a}{\left(\Omega_{0, m} a+\Omega_{0, \Lambda} a^{4}+\Omega_{0, r}\right)^{1 / 2}} \mathrm{~d} a $$ If $\Omega_{0, r}=0$ and $\Omega_{0, m}+\Omega_{0, \Lambda}=1$ (corresponding to what it known as a flat Universe), then via the standard integral $\int\left(b^{2}+x^{2}\right)^{-1 / 2} \mathrm{~d} x=\ln \left(x+\sqrt{b^{2}+x^{2}}\right)+C$ this integral can be evaluated analytically to give $$ t=t_{\mathrm{H}_{0}} \frac{2}{3 \Omega_{0, \Lambda}^{1 / 2}} \ln \left[\left(\frac{\Omega_{0, \Lambda}}{\Omega_{0, m}}\right)^{1 / 2}(1+z)^{-3 / 2}+\left(\frac{\Omega_{0, \Lambda}}{\Omega_{0, m}(1+z)^{3}}+1\right)^{1 / 2}\right] $$ Finally, the luminosity distance, $D_{L}(z)$, corresponding to the distance away that an object appears to be due to its measured flux given its intrinsic luminosity (i.e. $f \equiv L / 4 \pi D_{L}^{2}$ ) is given as $$ D_{L}=\left(1+z_{i}\right) D_{\mathrm{H}_{0}} \int_{0}^{z_{i}} \frac{1}{E(z)} \mathrm{d} z=\left(1+z_{i}\right) D_{\mathrm{H}_{0}} \int_{a_{i}}^{1} \frac{1}{\left(\Omega_{0, m} a+\Omega_{0, \Lambda} a^{4}+\Omega_{0, r}\right)^{1 / 2}} \mathrm{~d} a $$ where $z_{i}$ is the redshift of interest and $a_{i}$ is the equivalent scale factor. Even for the flat Universe case with $\Omega_{0, r}=0$ this integral cannot be be done analytically so must be evaluated numerically.a. The telescope will spend its expected 10-20 year mission in a halo orbit about the second Lagrangian point, L2 (see Figure 5). This is one of five special points in the Sun-Earth system where the gravitational forces from the two bodies provide the centripetal force required to have a (small mass) object there have an orbital period identical to the Earth. At the L2 point, this means it orbits quicker than you would expect for an object that distance from the Sun. i. Taking 1 year as 365.25 days and 1 au as $1.496 \times 10^{11} \mathrm{~m}$, using numerical methods show that the distance between the Earth and L2 is $~ 1.5 \times 10^{6} \mathrm{~km}$. Give your answer to 4 s.f.

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 James Webb Space Telescope (JWST) is an incredibly exciting next generation telescope that was successfully launched on $25^{\text {th }}$ December 2021 . Its mirror is approximately $6.5 \mathrm{~m}$ in diameter, much larger than the $2.4 \mathrm{~m}$ mirror of the Hubble Space Telescope (HST), and so it has far greater resolution and sensitivity. Whilst HST largely imaged in the visible, JWST will do most of its work in the nearand mid-infrared (NIR and MIR respectively). This will allow it to pick up heavily redshifted light, such as that from the first generation of stars in the very first galaxies. [figure1] Figure 5: Left: A full-scale model of JWST next to some of the scientists and engineers involved in its development at the Goddard Space Flight Center. Credit: NASA / Goddard Space Flight Center / Pat Izzo. Right: The position of the second Lagrangian point, $L_{2}$, relative to the Earth. Credit: ESA. The resolution limit of a telescope is set by the amount of diffraction light rays experience as they enter the system, and is related to the diameter of a telescope, $D$, and the wavelength being observed, $\lambda$. The resolution limit of a CCD is set by the size of the pixels. Three of the imaging cameras on JWST are tabulated with some properties below: | Instrument | Wavelength range $(\mu \mathrm{m})$ | CCD plate scale (arcseconds / pixel) | | :---: | :---: | :---: | | NIRCam (short wave) | $0.6-2.3$ | 0.031 | | NIRCam (long wave) | $2.4-5.0$ | 0.065 | | MIRI | $5.6-25.5$ | 0.11 | An arcsecond is a measure of angle where $1^{\circ}=3600$ arcseconds. The familiar variation in intensity on a screen, $I_{\text {slit }}$, due to diffraction through an infinitely tall single slit is given as $$ I_{\text {slit }}=I_{0}\left(\frac{\sin (x)}{x}\right)^{2}, \text { where } \quad x=\frac{\pi D \theta}{\lambda} $$ and $I_{0}$ is the initial intensity. For a circular aperture, the formula is slightly different and is given as $$ I_{\mathrm{circ}}=I_{0}\left(\frac{2 J_{1}(x)}{x}\right)^{2} . $$ Here $J_{1}(x)$ is the Bessel function of the first kind and is calculated as $$ J_{n}(x)=\sum_{r=0}^{\infty} \frac{(-1)^{r}}{r !(n+r) !}\left(\frac{x}{2}\right)^{n+2 r} \quad \text { so } \quad J_{1}(x)=\frac{x}{2}\left(1-\frac{x^{2}}{8}+\frac{x^{4}}{192}-\ldots\right) . $$ The $x$-axis intercepts and shape of the maxima are quite different, as shown in Figure 6. The position of the first minimum of $I_{\text {slit }}$ is at $x_{\min }=\pi$ meaning that $\theta_{\min , \text { slit }}=\lambda / D$, whilst for $I_{\text {circ }}$ it is at $x_{\min }=3.8317 \ldots$ so $\theta_{\min , \mathrm{circ}} \approx 1.22 \lambda / D$. This is one way of defining the minimum angular resolution, although since the flux drops off so steeply away from the central maximum a more convenient one for use with CCDs is the angle corresponding to the full width half maximum (FWHM). [figure2] Figure 6: Left: The $I_{\text {slit }}$ (purple) and $I_{\text {circ }}$ (blue - the wider central maximum) functions, normalised so that $I_{0}=1$. You can see the shapes and $x$-intercepts are different. Right: How $x_{\min }$ and the full width half maximum (FWHM) are defined. Here it is shown for $I_{\text {circ }}$. As well as having the largest mirror of any space telescope ever launched, it is also one of the most sensitive, with its greatest sensitivity in the NIRCam F200W filter (centred on a wavelength of $1.989 \mu \mathrm{m})$ where after $10^{4}$ seconds it can detect a flux of $9.1 \mathrm{nJy}\left(1 \mathrm{Jy}=10^{-26} \mathrm{~W} \mathrm{~m}^{-2} \mathrm{~Hz}^{-1}\right.$ ) with a signal-to-noise ratio (S/N) of 10 , corresponding to an apparent magnitude of $m=29.0$. This extraordinary sensitivity can be used to pick up light from the earliest galaxies in the Universe. The scale factor, $a$, parameterises the expansion of the Universe since the Big Bang, and is related to the redshift, $z$, as $$ a=(1+z)^{-1} \quad \text { where } \quad z \equiv \frac{\lambda_{\text {obs }}-\lambda_{\mathrm{emit}}}{\lambda_{\mathrm{emit}}} $$ with $\lambda_{\text {obs }}$ the observed wavelength and $\lambda_{\text {emit }}$ the rest frame wavelength. The current rate of expansion of the Universe is given by the Hubble constant, $H_{0}$, and this is related to the current Hubble time, $t_{\mathrm{H}_{0}}$, and current Hubble distance, $D_{\mathrm{H}_{0}}$, as $$ t_{\mathrm{H}_{0}} \equiv H_{0}^{-1} \quad \text { and } \quad D_{\mathrm{H}_{0}} \equiv c t_{\mathrm{H}_{0}} \text {. } $$ Here the subscript 0 indicates the values are measured today. The Hubble constant is more appropriately known as the Hubble parameter as it is a function of time, and the evolution of $H$ as a function of $z$ is $$ E(z)=\frac{H}{H_{0}} \equiv\left[\Omega_{0, m}(1+z)^{3}+\Omega_{0, \Lambda}+\Omega_{0, r}(1+z)^{4}\right]^{1 / 2}, $$ where $\Omega$ is the normalised density parameter, and the subscript $m, r$, and $\Lambda$ indicate the contribution to $\Omega$ from matter, radiation, and dark energy, respectively. The proper age of the Universe $t(z)$ at redshift $z$ is best evaluated in terms of $a$ as $$ t=t_{\mathrm{H}_{0}} \int_{0}^{(1+z)^{-1}} \frac{a}{\left(\Omega_{0, m} a+\Omega_{0, \Lambda} a^{4}+\Omega_{0, r}\right)^{1 / 2}} \mathrm{~d} a $$ If $\Omega_{0, r}=0$ and $\Omega_{0, m}+\Omega_{0, \Lambda}=1$ (corresponding to what it known as a flat Universe), then via the standard integral $\int\left(b^{2}+x^{2}\right)^{-1 / 2} \mathrm{~d} x=\ln \left(x+\sqrt{b^{2}+x^{2}}\right)+C$ this integral can be evaluated analytically to give $$ t=t_{\mathrm{H}_{0}} \frac{2}{3 \Omega_{0, \Lambda}^{1 / 2}} \ln \left[\left(\frac{\Omega_{0, \Lambda}}{\Omega_{0, m}}\right)^{1 / 2}(1+z)^{-3 / 2}+\left(\frac{\Omega_{0, \Lambda}}{\Omega_{0, m}(1+z)^{3}}+1\right)^{1 / 2}\right] $$ Finally, the luminosity distance, $D_{L}(z)$, corresponding to the distance away that an object appears to be due to its measured flux given its intrinsic luminosity (i.e. $f \equiv L / 4 \pi D_{L}^{2}$ ) is given as $$ D_{L}=\left(1+z_{i}\right) D_{\mathrm{H}_{0}} \int_{0}^{z_{i}} \frac{1}{E(z)} \mathrm{d} z=\left(1+z_{i}\right) D_{\mathrm{H}_{0}} \int_{a_{i}}^{1} \frac{1}{\left(\Omega_{0, m} a+\Omega_{0, \Lambda} a^{4}+\Omega_{0, r}\right)^{1 / 2}} \mathrm{~d} a $$ where $z_{i}$ is the redshift of interest and $a_{i}$ is the equivalent scale factor. Even for the flat Universe case with $\Omega_{0, r}=0$ this integral cannot be be done analytically so must be evaluated numerically. problem: a. The telescope will spend its expected 10-20 year mission in a halo orbit about the second Lagrangian point, L2 (see Figure 5). This is one of five special points in the Sun-Earth system where the gravitational forces from the two bodies provide the centripetal force required to have a (small mass) object there have an orbital period identical to the Earth. At the L2 point, this means it orbits quicker than you would expect for an object that distance from the Sun. i. Taking 1 year as 365.25 days and 1 au as $1.496 \times 10^{11} \mathrm{~m}$, using numerical methods show that the distance between the Earth and L2 is $~ 1.5 \times 10^{6} \mathrm{~km}$. Give your answer to 4 s.f. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of \mathrm{~m}, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

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Astronomy_272

按照我国整个月球探测活动的计划, 在第一步“绕月”工程圆满完成各项目标和科学探测任务后, 第二步是“落月”工程。已在 2013 年以前完成。假设月球半径为 $R$, 月球表面的重力加速度为 $g_{0}$, 飞船沿距月球表面高度为 $3 R$ 的圆形轨道I运动, 到达轨道的 $A$点时点火变轨进入椭圆轨道II, 到达轨道的近月点 $B$ 时再次点火进入月球近月轨道III绕月球做圆周运动。下列判断正确的是（） [图1] A: 飞船在轨道 $\mathrm{I}$ 上的运行速率 $v=\frac{\sqrt{g_{0} R}}{2}$ B: 飞船在 $A$ 点处点火变轨时, 速度增大 C: 飞船从 $A$ 到 $B$ 运行的过程中加速度增大 D: 飞船在轨道III绕月球运动一周所需的时间 $T=2 \pi \sqrt{\frac{R}{g_{0}}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 按照我国整个月球探测活动的计划, 在第一步“绕月”工程圆满完成各项目标和科学探测任务后, 第二步是“落月”工程。已在 2013 年以前完成。假设月球半径为 $R$, 月球表面的重力加速度为 $g_{0}$, 飞船沿距月球表面高度为 $3 R$ 的圆形轨道I运动, 到达轨道的 $A$点时点火变轨进入椭圆轨道II, 到达轨道的近月点 $B$ 时再次点火进入月球近月轨道III绕月球做圆周运动。下列判断正确的是（） [图1] A: 飞船在轨道 $\mathrm{I}$ 上的运行速率 $v=\frac{\sqrt{g_{0} R}}{2}$ B: 飞船在 $A$ 点处点火变轨时, 速度增大 C: 飞船从 $A$ 到 $B$ 运行的过程中加速度增大 D: 飞船在轨道III绕月球运动一周所需的时间 $T=2 \pi \sqrt{\frac{R}{g_{0}}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

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Astronomy

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multi-modal

Astronomy_95

根据图片中新闻信息, 并查阅资料获知火星的质量和万有引力常量, 则可估算出 ## 中国首次火星探测任务环绕火星获得成功新京报 据央视新闻客户端消息记者从国家航天局获悉, 刚刚，中国首次火星探测任务天问一号探测器实施近火捕获制动, 环绕器 $3000 \mathrm{~N}$ 轨控发动机点火工作约 15 分钟, 探测器顺利进入近火点高度约 400 千米, 周期约 10 个地球日, 倾角约 10 的大椭圆环火轨道, 成为我国第一颗人造火星卫星, 实现“绕、着、巡“第一步“绕”的目标, 环绕火星获得成功。 A: 火星的平均密度 B: 探测器的环火轨道半长轴 C: 火星表面的重力加速度 D: 探测器在近火点的加速度

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: 根据图片中新闻信息, 并查阅资料获知火星的质量和万有引力常量, 则可估算出 ## 中国首次火星探测任务环绕火星获得成功新京报 据央视新闻客户端消息记者从国家航天局获悉, 刚刚，中国首次火星探测任务天问一号探测器实施近火捕获制动, 环绕器 $3000 \mathrm{~N}$ 轨控发动机点火工作约 15 分钟, 探测器顺利进入近火点高度约 400 千米, 周期约 10 个地球日, 倾角约 10 的大椭圆环火轨道, 成为我国第一颗人造火星卫星, 实现“绕、着、巡“第一步“绕”的目标, 环绕火星获得成功。 A: 火星的平均密度 B: 探测器的环火轨道半长轴 C: 火星表面的重力加速度 D: 探测器在近火点的加速度 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]

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Astronomy

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text-only

Astronomy_427

金星是从太阳向外的第二颗行星, 假设金星和地球都是围绕太阳做匀速圆周运动,现从地球发射一颗金星探测器, 可以简化为这样的过程, 选择恰当的时间窗口, 探测器先脱离地球束缚成为和地球同轨道的人造小行星, 然后通过速度调整进入制圆转移轨道,经粗圆转移轨道 (关闭发动机) 到达金星轨道, 椭圆长轴的两端一端和地球轨道相切,一端和金星轨道相切。若太阳质量为 $M$, 探测器质量为 $m$, 太阳与探测器间距离为 $r$, 则它们之间的引力势能公式为 $E_{\mathrm{p}}=-\frac{G M m}{r}$ 。已知椭圆转移轨道与两圆轨道相切于 $A 、 B$两点且恰好对应椭圆的长轴, 地球轨道半径为 $r_{\mathrm{A}}$ 、周期为 $T_{\mathrm{A}}$, 金星轨道半径为 $r_{\mathrm{B}}$, 周期为 $T_{\mathrm{B}}$, 万有引力常量 $G$ 。忽略除太阳外其它星体对探测器的影响, 则 ( ) [图1] 地球轨道 A: 探测器在地球轨道上的 $A$ 点和转移轨道上的 $A$ 点处向心加速度不同 B: 探测器从地球轨道经转移轨道到达金星轨道的最短时间为 $\frac{T_{\mathrm{A}}+T_{\mathrm{B}}}{2}$ C: 探测器在转移轨道 $A$ 点的速度 $v_{\mathrm{A}}=\sqrt{\frac{2 r_{\mathrm{B}} G M}{r_{\mathrm{A}}\left(r_{\mathrm{A}}+r_{\mathrm{B}}\right)}}$ D: 探测器在地球正圆轨道的机械能比在金星正圆轨道的机械能小

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: 金星是从太阳向外的第二颗行星, 假设金星和地球都是围绕太阳做匀速圆周运动,现从地球发射一颗金星探测器, 可以简化为这样的过程, 选择恰当的时间窗口, 探测器先脱离地球束缚成为和地球同轨道的人造小行星, 然后通过速度调整进入制圆转移轨道,经粗圆转移轨道 (关闭发动机) 到达金星轨道, 椭圆长轴的两端一端和地球轨道相切,一端和金星轨道相切。若太阳质量为 $M$, 探测器质量为 $m$, 太阳与探测器间距离为 $r$, 则它们之间的引力势能公式为 $E_{\mathrm{p}}=-\frac{G M m}{r}$ 。已知椭圆转移轨道与两圆轨道相切于 $A 、 B$两点且恰好对应椭圆的长轴, 地球轨道半径为 $r_{\mathrm{A}}$ 、周期为 $T_{\mathrm{A}}$, 金星轨道半径为 $r_{\mathrm{B}}$, 周期为 $T_{\mathrm{B}}$, 万有引力常量 $G$ 。忽略除太阳外其它星体对探测器的影响, 则 ( ) [图1] 地球轨道 A: 探测器在地球轨道上的 $A$ 点和转移轨道上的 $A$ 点处向心加速度不同 B: 探测器从地球轨道经转移轨道到达金星轨道的最短时间为 $\frac{T_{\mathrm{A}}+T_{\mathrm{B}}}{2}$ C: 探测器在转移轨道 $A$ 点的速度 $v_{\mathrm{A}}=\sqrt{\frac{2 r_{\mathrm{B}} G M}{r_{\mathrm{A}}\left(r_{\mathrm{A}}+r_{\mathrm{B}}\right)}}$ D: 探测器在地球正圆轨道的机械能比在金星正圆轨道的机械能小 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

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multi-modal

Astronomy_1028

A typical cheap handheld telescope has a diameter of $10 \mathrm{~cm}$, whilst ones for keen amateurs can have diameters of $40 \mathrm{~cm}$. How much greater light gathering power does the larger telescope have? A: 2 B: 4 C: 8 D: 16

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 typical cheap handheld telescope has a diameter of $10 \mathrm{~cm}$, whilst ones for keen amateurs can have diameters of $40 \mathrm{~cm}$. How much greater light gathering power does the larger telescope have? A: 2 B: 4 C: 8 D: 16 You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

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Astronomy

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Astronomy_685

总质量为 $m$ 的返回式人造地球卫星沿半径为 $R$ 的圆轨道绕地球运动到某点时, 向原来运动方向喷出气体以降低卫星的速度, 随后卫星转到与地球相切的椭圆轨道, 要使卫星相对地面的速度变为原来的 $k$ 倍 $(k<1)$, 则卫星在该点将质量为 $\Delta m$ 的气体喷出的对地速度大小应为 (将连续喷气等效为一次性喷气, 地球半径为 $R_{0}$, 地球表面重力加速度为 $g$ ) A: $k \sqrt{\frac{g R_{0}^{2}}{R}}$ B: $\frac{1}{k} \sqrt{\frac{g R_{0}^{2}}{R}}$ C: $\left(\frac{m+k \Delta m-k m}{\Delta m}\right) \sqrt{\frac{g R_{0}^{2}}{R}}$ D: $\left(\frac{m-k \Delta m+k m}{\Delta m}\right) \sqrt{\frac{g R_{0}^{2}}{R}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 总质量为 $m$ 的返回式人造地球卫星沿半径为 $R$ 的圆轨道绕地球运动到某点时, 向原来运动方向喷出气体以降低卫星的速度, 随后卫星转到与地球相切的椭圆轨道, 要使卫星相对地面的速度变为原来的 $k$ 倍 $(k<1)$, 则卫星在该点将质量为 $\Delta m$ 的气体喷出的对地速度大小应为 (将连续喷气等效为一次性喷气, 地球半径为 $R_{0}$, 地球表面重力加速度为 $g$ ) A: $k \sqrt{\frac{g R_{0}^{2}}{R}}$ B: $\frac{1}{k} \sqrt{\frac{g R_{0}^{2}}{R}}$ C: $\left(\frac{m+k \Delta m-k m}{\Delta m}\right) \sqrt{\frac{g R_{0}^{2}}{R}}$ D: $\left(\frac{m-k \Delta m+k m}{\Delta m}\right) \sqrt{\frac{g R_{0}^{2}}{R}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

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text-only

Astronomy_867

In 1974, Stephen Hawking proved that black holes emit blackbody radiation according to the Stefan-Boltzmann law (due to quantum effects near the event horizon). This radiation is called Hawking radiation and through this process, black holes slowly evaporate their mass away in the absence of new material to accrete. Assume that the Hawking temperature of a black hole is inversely proportional to its mass (i.e. $T_{H}=$ const. / M) and that our initial black hole of mass $\mathrm{M}$ gets split into $\mathrm{N}$ smaller black holes, each with a mass $M$ / $N$. Using the results found in problems 6 and 7, what is the relation between the final combined luminosity of the smaller black holes $(\mathrm{L})$ and the luminosity of the initial black hole $\left(L_{0}\right)$ ? A: $L=L_{0}$ B: $L=L_{0} / N$ C: $L=N L_{0}$ D: $L=N^{2} L_{0}$ E: $L=N^{3} L_{0}$

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 1974, Stephen Hawking proved that black holes emit blackbody radiation according to the Stefan-Boltzmann law (due to quantum effects near the event horizon). This radiation is called Hawking radiation and through this process, black holes slowly evaporate their mass away in the absence of new material to accrete. Assume that the Hawking temperature of a black hole is inversely proportional to its mass (i.e. $T_{H}=$ const. / M) and that our initial black hole of mass $\mathrm{M}$ gets split into $\mathrm{N}$ smaller black holes, each with a mass $M$ / $N$. Using the results found in problems 6 and 7, what is the relation between the final combined luminosity of the smaller black holes $(\mathrm{L})$ and the luminosity of the initial black hole $\left(L_{0}\right)$ ? A: $L=L_{0}$ B: $L=L_{0} / N$ C: $L=N L_{0}$ D: $L=N^{2} L_{0}$ E: $L=N^{3} L_{0}$ 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_139

引力波探测于 2017 年获得诺贝尔物理学奖。双星的运动是引力波的来源之一，假设宇宙中有一双星系统由 $\mathrm{P} 、 \mathrm{Q}$ 两颗星体组成, 这两颗星绕它们连线上的某一点在二者之间万有引力作用下做匀速圆周运动, 测得 $\mathrm{P}$ 星的周期为 $T, \mathrm{P} 、 \mathrm{Q}$ 两颗星之间的距离为 $l, \mathrm{P} 、 \mathrm{Q}$ 两颗星的轨道半径之差为 $\Delta r$ ( $\mathrm{P}$ 星的轨道半径大于 $\mathrm{Q}$ 星的轨道半径), 引力常量为 $G$, 则 $(\quad)$ A: $\mathrm{P} 、 \mathrm{Q}$ 两颗星的向心力大小相等 B: $\mathrm{P} 、 \mathrm{Q}$ 两颗星的线速度之差为 $\frac{2 \pi \Delta r}{G T^{2}}$ C: $\mathrm{P} 、 \mathrm{Q}$ 两颗星的质量之差为 $\frac{4 \pi^{2} l^{2} \Delta r}{G T^{2}}$ D: $\mathrm{P} 、 \mathrm{Q}$ 两颗星的质量之和为 $\frac{4 \pi^{2} l^{2}}{G T^{2}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 引力波探测于 2017 年获得诺贝尔物理学奖。双星的运动是引力波的来源之一，假设宇宙中有一双星系统由 $\mathrm{P} 、 \mathrm{Q}$ 两颗星体组成, 这两颗星绕它们连线上的某一点在二者之间万有引力作用下做匀速圆周运动, 测得 $\mathrm{P}$ 星的周期为 $T, \mathrm{P} 、 \mathrm{Q}$ 两颗星之间的距离为 $l, \mathrm{P} 、 \mathrm{Q}$ 两颗星的轨道半径之差为 $\Delta r$ ( $\mathrm{P}$ 星的轨道半径大于 $\mathrm{Q}$ 星的轨道半径), 引力常量为 $G$, 则 $(\quad)$ A: $\mathrm{P} 、 \mathrm{Q}$ 两颗星的向心力大小相等 B: $\mathrm{P} 、 \mathrm{Q}$ 两颗星的线速度之差为 $\frac{2 \pi \Delta r}{G T^{2}}$ C: $\mathrm{P} 、 \mathrm{Q}$ 两颗星的质量之差为 $\frac{4 \pi^{2} l^{2} \Delta r}{G T^{2}}$ D: $\mathrm{P} 、 \mathrm{Q}$ 两颗星的质量之和为 $\frac{4 \pi^{2} l^{2}}{G T^{2}}$ 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应为两个或更多的选项：[A, B, C, D]

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Astronomy

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text-only

Astronomy_1044

Which of the following is not a zodiacal constellation? A: Virgo B: Cancer C: Aquila D: Gemini

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 not a zodiacal constellation? A: Virgo B: Cancer C: Aquila D: Gemini You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

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Astronomy

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Astronomy_802

Consider a satellite that has a circular orbit with a radius of $6.0 \times 10^{8} \mathrm{~m}$ around Venus. Due to a failure in its ignition system, the satellite's orbital velocity was suddenly decreased to zero during a maneuver. How long does the satellite take to hit the surface of the planet? Consider that the mass of Venus is $4.67 \times 10^{24} \mathrm{~kg}$ and neglect any gravitational effects on the satellite other than that from Venus. A: 15 hours. B: 3 days. C: 11 days. D: 25 days. E: 37 days.

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: Consider a satellite that has a circular orbit with a radius of $6.0 \times 10^{8} \mathrm{~m}$ around Venus. Due to a failure in its ignition system, the satellite's orbital velocity was suddenly decreased to zero during a maneuver. How long does the satellite take to hit the surface of the planet? Consider that the mass of Venus is $4.67 \times 10^{24} \mathrm{~kg}$ and neglect any gravitational effects on the satellite other than that from Venus. A: 15 hours. B: 3 days. C: 11 days. D: 25 days. E: 37 days. You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D, E].

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Astronomy

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text-only

Astronomy_232

太阳系中各行星几乎在同一平面内沿同一方向绕太阳做圆周运动。“行星冲日”是指某行星、地球和太阳几乎排成一直线的状态，地球位于太阳与该行星之间。已知相邻两次“冲日”的时间间隔火星约为 800 天，土星约为 378 天，则（） A: 火星公转周期约为 1.8 年 B: 火星的公转周期比土星的公转周期大 C: 火星的公转轨道半径比土星的公转轨道半径大 D: 火星和土星的公转轨道半径之比为 $\sqrt[3]{\left(\frac{800}{378}\right)^{2}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 太阳系中各行星几乎在同一平面内沿同一方向绕太阳做圆周运动。“行星冲日”是指某行星、地球和太阳几乎排成一直线的状态，地球位于太阳与该行星之间。已知相邻两次“冲日”的时间间隔火星约为 800 天，土星约为 378 天，则（） A: 火星公转周期约为 1.8 年 B: 火星的公转周期比土星的公转周期大 C: 火星的公转轨道半径比土星的公转轨道半径大 D: 火星和土星的公转轨道半径之比为 $\sqrt[3]{\left(\frac{800}{378}\right)^{2}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

text-only

Astronomy_1130

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.b. Determine the relationship between $\beta$ and $\theta$ that maximises $\beta_{\text {app }}$ for a given value of $\beta$, and hence determine the minimum value of $\beta$ needed to give rise to superluminal apparent speeds (i.e. when $\beta_{a p p}^{\max }>1$ ).

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. 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: b. Determine the relationship between $\beta$ and $\theta$ that maximises $\beta_{\text {app }}$ for a given value of $\beta$, and hence determine the minimum value of $\beta$ needed to give rise to superluminal apparent speeds (i.e. when $\beta_{a p p}^{\max }>1$ ). 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

EN

multi-modal

Astronomy_84

两颗卫星在同一轨道平面绕地球做匀速圆周运动, 地球半径为 $R, a$ 卫星离地面的高度等于 $R, b$ 卫星离地面高度为 $3 R$, 则: (2) 若某时刻两卫星正好同时通过地面同一点的正上方, 则 $a$ 至少经过多少个周期两卫星相距最远？

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题有多个正确答案，你需要包含所有。 问题: 两颗卫星在同一轨道平面绕地球做匀速圆周运动, 地球半径为 $R, a$ 卫星离地面的高度等于 $R, b$ 卫星离地面高度为 $3 R$, 则: (2) 若某时刻两卫星正好同时通过地面同一点的正上方, 则 $a$ 至少经过多少个周期两卫星相距最远？ 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。 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

MA

Astronomy

ZH

text-only

Astronomy_1170

In order to move a spacecraft between orbits we must apply a thrust using rockets, which changes the velocity of the spacecraft by $\Delta v$. In this question we will ignore changes in the mass of the spacecraft due to the burning of fuel. For an object of mass $m$ in a circular orbit of radius $r$ around an object with mass $M$ (where $m \ll M$ ) the orbital velocity, $v_{\text {orb }}$, is given by the formula $v_{\text {orb }}=\sqrt{\frac{G M}As part of their plan to rule the galaxy the First Order has created the Starkiller Base. Built within an ice planet and with a superweapon capable of destroying entire star systems, it is charged using the power of stars. The Starkiller Base has moved into the solar system and seeks to use the Sun to power its weapon to destroy the Earth. [figure1] Figure 3: The Starkiller Base charging its superweapon by draining energy from the local star. Credit: Star Wars: The Force Awakens, Lucasfilm. For this question you will need that the gravitational binding energy, $U$, of a uniform density spherical object with mass $M$ and radius $R$ is given by $$ U=\frac{3 G M^{2}}{5 R} $$ and that the mass-luminosity relation of low-mass main sequence stars is given by $L \propto M^{4}$.{r}}$.e. The Starkiller Base wants to destroy all the planets in a stellar system on the far side of the galaxy and so drains $0.10 M_{\odot}$ from the Sun to charge its weapon. Assuming that the $U$ per unit volume of the Sun stays approximately constant during this process, calculate: iii. The new temperature of the surface of the Sun (current $T_{\odot}=5780 \mathrm{~K}$ ), and suggest (with a suitable calculation) what change will be seen in terms of its colour.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: In order to move a spacecraft between orbits we must apply a thrust using rockets, which changes the velocity of the spacecraft by $\Delta v$. In this question we will ignore changes in the mass of the spacecraft due to the burning of fuel. For an object of mass $m$ in a circular orbit of radius $r$ around an object with mass $M$ (where $m \ll M$ ) the orbital velocity, $v_{\text {orb }}$, is given by the formula $v_{\text {orb }}=\sqrt{\frac{G M}As part of their plan to rule the galaxy the First Order has created the Starkiller Base. Built within an ice planet and with a superweapon capable of destroying entire star systems, it is charged using the power of stars. The Starkiller Base has moved into the solar system and seeks to use the Sun to power its weapon to destroy the Earth. [figure1] Figure 3: The Starkiller Base charging its superweapon by draining energy from the local star. Credit: Star Wars: The Force Awakens, Lucasfilm. For this question you will need that the gravitational binding energy, $U$, of a uniform density spherical object with mass $M$ and radius $R$ is given by $$ U=\frac{3 G M^{2}}{5 R} $$ and that the mass-luminosity relation of low-mass main sequence stars is given by $L \propto M^{4}$.{r}}$. problem: e. The Starkiller Base wants to destroy all the planets in a stellar system on the far side of the galaxy and so drains $0.10 M_{\odot}$ from the Sun to charge its weapon. Assuming that the $U$ per unit volume of the Sun stays approximately constant during this process, calculate: iii. The new temperature of the surface of the Sun (current $T_{\odot}=5780 \mathrm{~K}$ ), and suggest (with a suitable calculation) what change will be seen in terms of its colour. 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_184

拉格朗日点指在两个大天体引力作用下, 能使小物体稳定的点 (小物体质量相对两 大天体可忽略不计)。这些点的存在由法国数学家拉格朗日于 1772 年推导证明的, 1906 年首次发现运动于木星轨道上的小行星 (见脱罗央群小行星) 在木星和太阳的作用下处于拉格朗日点上。在每个由两大天体构成的系统中, 按推论有 5 个拉格朗日点, 其中连线上有三个拉格朗日点, 分别是 $L_{1} 、 L_{2} 、 L_{3}$, 如图所示。我国发射的“鹊桥”卫星就在地月系统平衡点 $L_{2}$ 点做周期运动, 通过定期轨控保持轨道的稳定性, 可实现对着陆器和巡视器的中继通信覆盖, 首次实现地月 $L_{2}$ 点周期轨道的长期稳定运行。设某两个天体系统的中心天体质量为 $M$, 环绕天体质量为 $m$, 两天体间距离为 $L$, 万有引力常量为 $G, L_{1}$ 点到中心天体的距离为 $R_{1}, L_{2}$ 点到中心天体的距离为 $R_{2}$ 。求: 处于 $L_{l}$ 点小物体的向心加速度; [图1]

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 拉格朗日点指在两个大天体引力作用下, 能使小物体稳定的点 (小物体质量相对两 大天体可忽略不计)。这些点的存在由法国数学家拉格朗日于 1772 年推导证明的, 1906 年首次发现运动于木星轨道上的小行星 (见脱罗央群小行星) 在木星和太阳的作用下处于拉格朗日点上。在每个由两大天体构成的系统中, 按推论有 5 个拉格朗日点, 其中连线上有三个拉格朗日点, 分别是 $L_{1} 、 L_{2} 、 L_{3}$, 如图所示。我国发射的“鹊桥”卫星就在地月系统平衡点 $L_{2}$ 点做周期运动, 通过定期轨控保持轨道的稳定性, 可实现对着陆器和巡视器的中继通信覆盖, 首次实现地月 $L_{2}$ 点周期轨道的长期稳定运行。设某两个天体系统的中心天体质量为 $M$, 环绕天体质量为 $m$, 两天体间距离为 $L$, 万有引力常量为 $G, L_{1}$ 点到中心天体的距离为 $R_{1}, L_{2}$ 点到中心天体的距离为 $R_{2}$ 。求: 处于 $L_{l}$ 点小物体的向心加速度; [图1] 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

EX

Astronomy

ZH

multi-modal

Astronomy_170

火星各种环境与地球十分相似, 人类对未来移居火星有着强烈的期望。地球体积为火星的 7 倍。质量为火星的 11 倍。假设某天人类移居火星后, 小华同学在火星表面制造了如下装置。如图所示。半径为 $r=1 \mathrm{~m}$ 的光滑圆弧固定在坚直平面内, 其末端与木板 $B$ 的上表面所在平面相切, 且初始时木板 $B$ 的左端刚好与圆弧末端对齐, 木板 $B$ 带电,电荷量为 $1 \mathrm{C}$, 木板 $B$ 左端紧挨着光滑小物块 $A$, 小物块 $A$ 左侧有一橡胶墙壁, 能与 $A$发生弹性正碰, 空间内存在水平向左的匀强电场, 电场强度 $E=1 \mathrm{~N} / \mathrm{C}$, 开始时由圆弧轨道上端静止释放一带电小物块 $C$, 电荷量 $q_{c}=-\frac{1}{3} C$, 当小物块 $C$ 达到圆弧最底端时,其对圆弧轨道的压力大小为 $\frac{28}{3} \mathrm{~N}$, 此时 $A B$ 之间存在的炸药爆炸, 给予 $A B$ 等量的动能, 动能为 $2 \mathrm{~J}, C$ 与 $B$ 之间的动摩擦因数为 $\mu_{1}=0.2, B$ 与水平面间的动摩擦因数为 $\mu_{2}=0.1$, $A C$ 质量未知。 $B$ 的质量 $m_{B}=4 \mathrm{~kg}$, 已知地球表面的重力加速度 $g=10 \mathrm{~m} / \mathrm{s}^{2}$ 。为简化计算取 $\frac{\sqrt[3]{49}}{11}=\frac{1}{3}$ 。 求 $C$ 的质量 [图1]

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个数值。 问题: 火星各种环境与地球十分相似, 人类对未来移居火星有着强烈的期望。地球体积为火星的 7 倍。质量为火星的 11 倍。假设某天人类移居火星后, 小华同学在火星表面制造了如下装置。如图所示。半径为 $r=1 \mathrm{~m}$ 的光滑圆弧固定在坚直平面内, 其末端与木板 $B$ 的上表面所在平面相切, 且初始时木板 $B$ 的左端刚好与圆弧末端对齐, 木板 $B$ 带电,电荷量为 $1 \mathrm{C}$, 木板 $B$ 左端紧挨着光滑小物块 $A$, 小物块 $A$ 左侧有一橡胶墙壁, 能与 $A$发生弹性正碰, 空间内存在水平向左的匀强电场, 电场强度 $E=1 \mathrm{~N} / \mathrm{C}$, 开始时由圆弧轨道上端静止释放一带电小物块 $C$, 电荷量 $q_{c}=-\frac{1}{3} C$, 当小物块 $C$ 达到圆弧最底端时,其对圆弧轨道的压力大小为 $\frac{28}{3} \mathrm{~N}$, 此时 $A B$ 之间存在的炸药爆炸, 给予 $A B$ 等量的动能, 动能为 $2 \mathrm{~J}, C$ 与 $B$ 之间的动摩擦因数为 $\mu_{1}=0.2, B$ 与水平面间的动摩擦因数为 $\mu_{2}=0.1$, $A C$ 质量未知。 $B$ 的质量 $m_{B}=4 \mathrm{~kg}$, 已知地球表面的重力加速度 $g=10 \mathrm{~m} / \mathrm{s}^{2}$ 。为简化计算取 $\frac{\sqrt[3]{49}}{11}=\frac{1}{3}$ 。 求 $C$ 的质量 [图1] 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 请记住，你的答案应以kg为单位计算，但在给出最终答案时，请不要包含单位。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。

NV

Astronomy

ZH

multi-modal

Astronomy_761

The astronomical unit parsec (pc) plays a crucial role in astronomy. One parsec is equal to about 3.26 light-years. How is one parsec defined in astronomy? A: Distance at which one astronomical unit measures one arcsecond from Earth. B: Orbital distance of the solar system around the center of the Milky Way in one year. C: Effective distance of the solar wind (i.e. the radius of the heliosphere). D: Historical distance to the brightest star Sirius.

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 astronomical unit parsec (pc) plays a crucial role in astronomy. One parsec is equal to about 3.26 light-years. How is one parsec defined in astronomy? A: Distance at which one astronomical unit measures one arcsecond from Earth. B: Orbital distance of the solar system around the center of the Milky Way in one year. C: Effective distance of the solar wind (i.e. the radius of the heliosphere). D: Historical distance to the brightest star Sirius. 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_240

2019 年 9 月 12 日, 我国在太原卫星发射中心又一次“一箭三星”发射成功。现假设三颗星 $a 、 b 、 c$ 均在赤道平面上空绕地球做匀速圆周运动, 其中 $a 、 b$ 转动方向与地球自转方向相同, $c$ 转动方向与地球自转方向相反, $a 、 b 、 c$ 三颗星的周期分别为 $T_{a}=6 \mathrm{~h}$ 、 $T_{b}=24 \mathrm{~h} 、 T_{c}=12 \mathrm{~h}$, 某一时刻三个卫星位置如图所示, 从该时刻起, 下列说法正确的是 [图1] A: $a 、 b$ 每经过 $4 \mathrm{~h}$ 相距最近一次 B: $a 、 b$ 经过 $8 \mathrm{~h}$ 第一次相距最远 C: $b 、 c$ 经过 $4 \mathrm{~h}$ 第一次相距最远 D: $b 、 c$ 每经过 $8 \mathrm{~h}$ 相距最近一次

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 2019 年 9 月 12 日, 我国在太原卫星发射中心又一次“一箭三星”发射成功。现假设三颗星 $a 、 b 、 c$ 均在赤道平面上空绕地球做匀速圆周运动, 其中 $a 、 b$ 转动方向与地球自转方向相同, $c$ 转动方向与地球自转方向相反, $a 、 b 、 c$ 三颗星的周期分别为 $T_{a}=6 \mathrm{~h}$ 、 $T_{b}=24 \mathrm{~h} 、 T_{c}=12 \mathrm{~h}$, 某一时刻三个卫星位置如图所示, 从该时刻起, 下列说法正确的是 [图1] A: $a 、 b$ 每经过 $4 \mathrm{~h}$ 相距最近一次 B: $a 、 b$ 经过 $8 \mathrm{~h}$ 第一次相距最远 C: $b 、 c$ 经过 $4 \mathrm{~h}$ 第一次相距最远 D: $b 、 c$ 每经过 $8 \mathrm{~h}$ 相距最近一次 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应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_522

《流浪地球 2 》中太空电梯非常吸引观众眼球。太空电梯通过超级缆绳连接地球赤道上的固定基地与配重空间站, 它们随地球以同步静止状态一起旋转，如图所示。图中配重空间站质量为 $m$, 比同步卫星更高, 距地面高达 $9 R$ 。若地球半径为 $R$, 自转周期为 $T$, 重力加速度为 $g$, 求: 通过缆绳连接的配重空间站线速度大小为多少; [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: 《流浪地球 2 》中太空电梯非常吸引观众眼球。太空电梯通过超级缆绳连接地球赤道上的固定基地与配重空间站, 它们随地球以同步静止状态一起旋转，如图所示。图中配重空间站质量为 $m$, 比同步卫星更高, 距地面高达 $9 R$ 。若地球半径为 $R$, 自转周期为 $T$, 重力加速度为 $g$, 求: 通过缆绳连接的配重空间站线速度大小为多少; [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_554

我国首个火星探测器—“年问一号”, 于 2020 年 7 月 23 日在海南文昌航天发射中心成功发射, 计划于 2021 年 5 月至 6 月择机实施火星着陆, 开展巡视探测, 如图为“天问一号” "环绕火星变轨示意图。已知地球质量为 $M$, 地球半径为 $R$, 地球表面重力加速度为 $g$; 火星的质量约为地球质量的 $\frac{1}{9}$, 半径约为地球半径的 $\frac{1}{2}$; 着陆器质量为 $m$ 。下列说法正确的是 ( ) [图1] A: “天问一号”探测器环绕火星运动的速度应大于 $11.2 \mathrm{~km} / \mathrm{s}$ B: “天问一号”在轨道II运行到 $Q$ 点的速度大于在圆轨道 $\mathrm{I}$ 运行的速度 C: 若轨道I为近火星圆轨道, 测得周期为 $T$, 则火星的密度约为 $\frac{3 \pi M}{T^{2} R^{2} g}$ D: 着陆器在火星表面所受重力约为 $\frac{m g}{4}$

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: 我国首个火星探测器—“年问一号”, 于 2020 年 7 月 23 日在海南文昌航天发射中心成功发射, 计划于 2021 年 5 月至 6 月择机实施火星着陆, 开展巡视探测, 如图为“天问一号” "环绕火星变轨示意图。已知地球质量为 $M$, 地球半径为 $R$, 地球表面重力加速度为 $g$; 火星的质量约为地球质量的 $\frac{1}{9}$, 半径约为地球半径的 $\frac{1}{2}$; 着陆器质量为 $m$ 。下列说法正确的是 ( ) [图1] A: “天问一号”探测器环绕火星运动的速度应大于 $11.2 \mathrm{~km} / \mathrm{s}$ B: “天问一号”在轨道II运行到 $Q$ 点的速度大于在圆轨道 $\mathrm{I}$ 运行的速度 C: 若轨道I为近火星圆轨道, 测得周期为 $T$, 则火星的密度约为 $\frac{3 \pi M}{T^{2} R^{2} g}$ D: 着陆器在火星表面所受重力约为 $\frac{m g}{4}$ 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

multi-modal

Astronomy_120

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 $R$, 忽略星球的自转, 求该星球的密度;

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 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 $R$, 忽略星球的自转, 求该星球的密度; 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 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_484

人造地球卫星与地心间距离为 $r$ 时, 取无穷远处为零势能点, 引力势能可以表示为 $E_{\mathrm{p}}=-\frac{G M m}{r}$, 其中 $G$ 为引力常量, $M$ 为地球质量, $m$ 为卫星质量。卫星原来在半径为 $r_{1}$的轨道上绕地球做匀速圆周运动, 由于稀薄空气等因素的影响, 飞行一段时间后其圆周运动的半径减小为 $r_{2}$ 。此过程中损失的机械能为 ( ) A: $\frac{G M m}{2}\left(\frac{1}{r_{2}}-\frac{1}{r_{1}}\right)$ B: $\frac{G M m}{2}\left(\frac{1}{r_{1}}-\frac{1}{r_{2}}\right)$ C: $\operatorname{GMm}\left(\frac{1}{r_{2}}-\frac{1}{r_{1}}\right)$ D: $G M m\left(\frac{1}{r_{1}}-\frac{1}{r_{2}^{2}}\right)$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 人造地球卫星与地心间距离为 $r$ 时, 取无穷远处为零势能点, 引力势能可以表示为 $E_{\mathrm{p}}=-\frac{G M m}{r}$, 其中 $G$ 为引力常量, $M$ 为地球质量, $m$ 为卫星质量。卫星原来在半径为 $r_{1}$的轨道上绕地球做匀速圆周运动, 由于稀薄空气等因素的影响, 飞行一段时间后其圆周运动的半径减小为 $r_{2}$ 。此过程中损失的机械能为 ( ) A: $\frac{G M m}{2}\left(\frac{1}{r_{2}}-\frac{1}{r_{1}}\right)$ B: $\frac{G M m}{2}\left(\frac{1}{r_{1}}-\frac{1}{r_{2}}\right)$ C: $\operatorname{GMm}\left(\frac{1}{r_{2}}-\frac{1}{r_{1}}\right)$ D: $G M m\left(\frac{1}{r_{1}}-\frac{1}{r_{2}^{2}}\right)$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

text-only

Astronomy_1078

On $21^{\text {st }}$ August 2017 the continental United States experienced a total solar eclipse. Dubbed the 'Great American Eclipse', it was estimated to be one of the most watched eclipses in history. ## Total Solar Eclipse of 2017 Aug 21 [figure1] Figure 3: The path of totality for the Great American Eclipse. The narrow dimension is its width. Credit: Fred Espenak, NASA's GSFC. The path of totality (where the Moon completely obscures the Sun) is shown in Figure 3, and the point of greatest eclipse ("GE"; where the path was widest since the axis of the cone of the Moon's shadow passed closest to the centre of the Earth) was near the village of Cerulean, Kentucky. The following data can be used for this question: - The angular radii of the Sun and the Moon (if observed from the centre of the Earth) at the moment of GE are $15^{\prime} 48.7^{\prime \prime}$ and $16^{\prime} 03.4^{\prime \prime}$, respectively, where the notation $x x^{\prime} y y . y^{\prime \prime}$ corresponds to $x x$ arcminutes and $y y . y$ arcseconds ( 60 arcminutes $=1$ degree, and 60 arcseconds $=1$ arcminute $)$ - The latitude and longitude of the location of GE are $36^{\circ} 58.0^{\prime} \mathrm{N}$ and $87^{\circ} 40.3^{\prime} \mathrm{W}$, respectively - Take the mean radii of the Sun, Earth and Moon to be respectively $R_{\odot}=695700 \mathrm{~km}, R_{\oplus}=$ $6371 \mathrm{~km}$ and $R_{\text {Moon }}=1737 \mathrm{~km}$, and a day to be 24 hours - Take the semi-major axes of the Sun-Earth and Earth-Moon systems to be $149600000 \mathrm{~km}$ and $384400 \mathrm{~km}$, respectively - As viewed from a location far above the North Pole, the Moon orbits in an anticlockwise direction around the Earth, and the Earth spins in an anticlockwise direction For an ellipse with semi-major axis $a$ it can be shown that the velocity $v$, at a distance $r$ from mass $M$, can be written as: $$ v^{2}=G M\left(\frac{2}{r}-\frac{1}{a}\right) $$ The point of greatest eclipse and greatest duration do not generally coincide, as a more elliptical shadow with a major axis aligned with the path of maximum totality (and thinner path width, equal to the minor axis) can compensate for the shadow moving faster at higher latitudes. For this eclipse the point of greatest duration ("GD") was at co-ordinates of $37^{\circ} 35^{\prime} \mathrm{N}$ latitude and $89^{\circ} 07^{\prime} \mathrm{W}$ longitude, reached about 4 minutes before GE, and where totality lasted $0.1 \mathrm{~s}$ longer than the value calculated in part $\mathrm{c}$. [figure2] Figure 4: The route of the Moon's shadow in the vicinity of the points of greatest duration (GD, near Carbondale) and greatest eclipse (GE, near Hopkinsville). Any places between the two limits on the path of totality will experience at least a very short period of totality - outside that region will only be a partial eclipse and the perpendicular distance between them is the path width. The longest duration of totality at that point of the shadow's journey is indicated as the path of maximum totality, which both GE and GD sit on. The closest part of that path to Carbondale is indicated as CP. Credit: Fred Espenak \& Google Maps.d. The town of Carbondale, Illinois, is the closest big town to the point of GD, with co-ordinates $37^{\circ} 44^{\prime} \mathrm{N}$ latitude and $89^{\circ} 13^{\prime} \mathrm{W}$ longitude. Assuming the path of maximum totality can be treated as linear as it passes through the region around GD and GE: ii. Calculate the distance (in $\mathrm{km}$ ) between Carbondale and CP.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: On $21^{\text {st }}$ August 2017 the continental United States experienced a total solar eclipse. Dubbed the 'Great American Eclipse', it was estimated to be one of the most watched eclipses in history. ## Total Solar Eclipse of 2017 Aug 21 [figure1] Figure 3: The path of totality for the Great American Eclipse. The narrow dimension is its width. Credit: Fred Espenak, NASA's GSFC. The path of totality (where the Moon completely obscures the Sun) is shown in Figure 3, and the point of greatest eclipse ("GE"; where the path was widest since the axis of the cone of the Moon's shadow passed closest to the centre of the Earth) was near the village of Cerulean, Kentucky. The following data can be used for this question: - The angular radii of the Sun and the Moon (if observed from the centre of the Earth) at the moment of GE are $15^{\prime} 48.7^{\prime \prime}$ and $16^{\prime} 03.4^{\prime \prime}$, respectively, where the notation $x x^{\prime} y y . y^{\prime \prime}$ corresponds to $x x$ arcminutes and $y y . y$ arcseconds ( 60 arcminutes $=1$ degree, and 60 arcseconds $=1$ arcminute $)$ - The latitude and longitude of the location of GE are $36^{\circ} 58.0^{\prime} \mathrm{N}$ and $87^{\circ} 40.3^{\prime} \mathrm{W}$, respectively - Take the mean radii of the Sun, Earth and Moon to be respectively $R_{\odot}=695700 \mathrm{~km}, R_{\oplus}=$ $6371 \mathrm{~km}$ and $R_{\text {Moon }}=1737 \mathrm{~km}$, and a day to be 24 hours - Take the semi-major axes of the Sun-Earth and Earth-Moon systems to be $149600000 \mathrm{~km}$ and $384400 \mathrm{~km}$, respectively - As viewed from a location far above the North Pole, the Moon orbits in an anticlockwise direction around the Earth, and the Earth spins in an anticlockwise direction For an ellipse with semi-major axis $a$ it can be shown that the velocity $v$, at a distance $r$ from mass $M$, can be written as: $$ v^{2}=G M\left(\frac{2}{r}-\frac{1}{a}\right) $$ The point of greatest eclipse and greatest duration do not generally coincide, as a more elliptical shadow with a major axis aligned with the path of maximum totality (and thinner path width, equal to the minor axis) can compensate for the shadow moving faster at higher latitudes. For this eclipse the point of greatest duration ("GD") was at co-ordinates of $37^{\circ} 35^{\prime} \mathrm{N}$ latitude and $89^{\circ} 07^{\prime} \mathrm{W}$ longitude, reached about 4 minutes before GE, and where totality lasted $0.1 \mathrm{~s}$ longer than the value calculated in part $\mathrm{c}$. [figure2] Figure 4: The route of the Moon's shadow in the vicinity of the points of greatest duration (GD, near Carbondale) and greatest eclipse (GE, near Hopkinsville). Any places between the two limits on the path of totality will experience at least a very short period of totality - outside that region will only be a partial eclipse and the perpendicular distance between them is the path width. The longest duration of totality at that point of the shadow's journey is indicated as the path of maximum totality, which both GE and GD sit on. The closest part of that path to Carbondale is indicated as CP. Credit: Fred Espenak \& Google Maps. problem: d. The town of Carbondale, Illinois, is the closest big town to the point of GD, with co-ordinates $37^{\circ} 44^{\prime} \mathrm{N}$ latitude and $89^{\circ} 13^{\prime} \mathrm{W}$ longitude. Assuming the path of maximum totality can be treated as linear as it passes through the region around GD and GE: ii. Calculate the distance (in $\mathrm{km}$ ) between Carbondale and CP. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of \mathrm{~km}, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

multi-modal

Astronomy_1103

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, $C R=\frac{f_{\text {star }}}{f_{\text {planet }}}$.c. Verify that the HST (which is diffraction limited since it's in space) would be sensitive enough to image the planet in the visible, but is unable to resolve it from its host star (take $\lambda=550 \mathrm{~nm}$ ).

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: 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: c. Verify that the HST (which is diffraction limited since it's in space) would be sensitive enough to image the planet in the visible, but is unable to resolve it from its host star (take $\lambda=550 \mathrm{~nm}$ ). 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 radians, 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_944

In the rotating reference frame where the Earth is stationary, an asteroid orbits the Sun in 3.5 years. What is the distance between the asteroid and the Sun? A: 1.25 au B: $2.08 \mathrm{au}$ C: $3.95 \mathrm{au}$ D: 6.54 au

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: In the rotating reference frame where the Earth is stationary, an asteroid orbits the Sun in 3.5 years. What is the distance between the asteroid and the Sun? A: 1.25 au B: $2.08 \mathrm{au}$ C: $3.95 \mathrm{au}$ D: 6.54 au You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

SC

Astronomy

EN

text-only

Astronomy_1017

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. Calculate the apparent magnitude of the JWST at $L_{2}$ due to the light reflected off its sunshield, given the apparent magnitude of the Sun is $m_{\odot}=-26.832$ as viewed from Earth. [Hint: you may wish to calculate the intensity of light that corresponds to an apparent magnitude of zero.]

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. Calculate the apparent magnitude of the JWST at $L_{2}$ due to the light reflected off its sunshield, given the apparent magnitude of the Sun is $m_{\odot}=-26.832$ as viewed from Earth. [Hint: you may wish to calculate the intensity of light that corresponds to an apparent magnitude of zero.] 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_764

Scientists detect no $\mathrm{CH}_{3} \mathrm{OH}$ and no $\mathrm{NH}_{3}$ in the atmosphere of a sub-Neptune planet. What type of surface does this planet probably have? A: Shallow surface B: Water oceans C: Dry surface D: Methane oceans

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: Scientists detect no $\mathrm{CH}_{3} \mathrm{OH}$ and no $\mathrm{NH}_{3}$ in the atmosphere of a sub-Neptune planet. What type of surface does this planet probably have? A: Shallow surface B: Water oceans C: Dry surface D: Methane oceans 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_326

2020 年 1 月 7 号, 通信技术试验卫星五号发射升空, 卫星发射时一般需要先到圆轨道 1 , 然后通过变轨进入圆轨道 2 。假设卫星在两圆轨道上速率之比 $v_{1}: v_{2}=5: 3$, 卫星质量不变, 则 $(\quad)$ [图1] A: 卫星通过椭圆轨道进入轨道 2 时应减速 B: 卫星在两圆轨道运行时的角速度大小之比 $\omega_{1}: \omega_{2}=125: 27$ C: 卫星在 1 轨道运行时和地球之间的万有引力不变 D: 卫星在两圆轨道运行时的动能之比 $E_{k 1}: E_{k 2}=9: 25$

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 1 月 7 号, 通信技术试验卫星五号发射升空, 卫星发射时一般需要先到圆轨道 1 , 然后通过变轨进入圆轨道 2 。假设卫星在两圆轨道上速率之比 $v_{1}: v_{2}=5: 3$, 卫星质量不变, 则 $(\quad)$ [图1] A: 卫星通过椭圆轨道进入轨道 2 时应减速 B: 卫星在两圆轨道运行时的角速度大小之比 $\omega_{1}: \omega_{2}=125: 27$ C: 卫星在 1 轨道运行时和地球之间的万有引力不变 D: 卫星在两圆轨道运行时的动能之比 $E_{k 1}: E_{k 2}=9: 25$ 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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100 多年前爱因斯坦预言了引力波存在, 2015 年科学家探测到黑洞合并引起的引力波。双星的运动是产生引力波的来源之一, 在宇宙中有一双星系统由 $\mathrm{P} 、 \mathrm{Q}$ 两颗星体组成, 这两颗星绕它们连线的某一点只在二者间的万有引力作用下做匀速圆周运动, 测得 $\mathrm{P}$ 星的周期为 $\mathrm{T}, \mathrm{P} 、 \mathrm{Q}$ 两颗星的距离为 $l, \mathrm{P} 、 \mathrm{Q}$ 两颗星的轨道半径之差为 $\Delta r(\mathrm{P}$ 星的轨道半径大于 $\mathrm{Q}$ 星的轨道半径), 引力常量为 $\mathrm{G}$, 则下列结论错误的是 ( ) A: $\mathrm{Q} 、 \mathrm{P}$ 两颗星的质量差为 $\frac{4 \pi^{2} l^{2} \Delta r}{G T^{2}}$ B: $\mathrm{P} 、 \mathrm{Q}$ 两颗星的线速度大小之差为 $\frac{2 \pi \Delta r}{T}$ C: $\mathrm{Q} 、 \mathrm{P}$ 两颗星的质量之比为 $\frac{l}{l-\Delta r}$ D: $\mathrm{P} 、 \mathrm{Q}$ 两颗星的运动半径之比为 $\frac{l+\Delta r}{l-\Delta r}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 100 多年前爱因斯坦预言了引力波存在, 2015 年科学家探测到黑洞合并引起的引力波。双星的运动是产生引力波的来源之一, 在宇宙中有一双星系统由 $\mathrm{P} 、 \mathrm{Q}$ 两颗星体组成, 这两颗星绕它们连线的某一点只在二者间的万有引力作用下做匀速圆周运动, 测得 $\mathrm{P}$ 星的周期为 $\mathrm{T}, \mathrm{P} 、 \mathrm{Q}$ 两颗星的距离为 $l, \mathrm{P} 、 \mathrm{Q}$ 两颗星的轨道半径之差为 $\Delta r(\mathrm{P}$ 星的轨道半径大于 $\mathrm{Q}$ 星的轨道半径), 引力常量为 $\mathrm{G}$, 则下列结论错误的是 ( ) A: $\mathrm{Q} 、 \mathrm{P}$ 两颗星的质量差为 $\frac{4 \pi^{2} l^{2} \Delta r}{G T^{2}}$ B: $\mathrm{P} 、 \mathrm{Q}$ 两颗星的线速度大小之差为 $\frac{2 \pi \Delta r}{T}$ C: $\mathrm{Q} 、 \mathrm{P}$ 两颗星的质量之比为 $\frac{l}{l-\Delta r}$ D: $\mathrm{P} 、 \mathrm{Q}$ 两颗星的运动半径之比为 $\frac{l+\Delta r}{l-\Delta r}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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利用水流和太阳能发电, 可以为人类提供清洁能源。已知太阳光垂直照射到地面上时的辐射功率 $P_{0}=1.0 \times 10^{3} \mathrm{~W} / \mathrm{m}^{2}$, 地球表面的重力加速度取 $g=10 \mathrm{~m} / \mathrm{s}^{2}$, 水的密度 $\rho=1.0 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ 。 (1)若利用太阳能发电, 需要发射一颗卫星到地球同步轨道上, 然后通过微波持续不断地将电能输送到地面, 这样就建成了宇宙太阳能发电站。已知地球同步轨道半径约为地球半径的 $2 \sqrt{11}$ 倍。 (2) 三峡水电站发电机输出的电压为 $18 \mathrm{kV}$ 。若采用 $500 \mathrm{kV}$ 直流电向某地区输电 $5.0 \times 10^{6} \mathrm{~kW}$, 要求输电线上损耗的功率不高于输送功率的 $5 \%$ 三峡水电站水库面积约 $S^{\prime}=1.0 \times 10^{9} \mathrm{~m}^{2}$, 平均流量 $Q=1.5 \times 10^{3} \mathrm{~m}^{3} / \mathrm{s}$, 水库水面与 发电机所在位置的平均高度差为 $h=100 \mathrm{~m}$, 并且在发电过程中水库水面高度保持不变。发电站将水的势能转化为电能的总效率 $\eta=60 \%$ 。在地球同步轨道上, 太阳光垂直照射时的辐射功率为 $10 P_{0}$ 。太阳能电池板将太阳能转化为电能的效率为 $20 \%$, 将电能输送到地面的过程要损失 50\%。若要使（1）中的宇宙太阳能发电站与三峡电站具有相同的发电能力, 同步卫星上太阳能电池板的面积至少为多大?

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个数值。 问题: 利用水流和太阳能发电, 可以为人类提供清洁能源。已知太阳光垂直照射到地面上时的辐射功率 $P_{0}=1.0 \times 10^{3} \mathrm{~W} / \mathrm{m}^{2}$, 地球表面的重力加速度取 $g=10 \mathrm{~m} / \mathrm{s}^{2}$, 水的密度 $\rho=1.0 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ 。 (1)若利用太阳能发电, 需要发射一颗卫星到地球同步轨道上, 然后通过微波持续不断地将电能输送到地面, 这样就建成了宇宙太阳能发电站。已知地球同步轨道半径约为地球半径的 $2 \sqrt{11}$ 倍。 (2) 三峡水电站发电机输出的电压为 $18 \mathrm{kV}$ 。若采用 $500 \mathrm{kV}$ 直流电向某地区输电 $5.0 \times 10^{6} \mathrm{~kW}$, 要求输电线上损耗的功率不高于输送功率的 $5 \%$ 三峡水电站水库面积约 $S^{\prime}=1.0 \times 10^{9} \mathrm{~m}^{2}$, 平均流量 $Q=1.5 \times 10^{3} \mathrm{~m}^{3} / \mathrm{s}$, 水库水面与 发电机所在位置的平均高度差为 $h=100 \mathrm{~m}$, 并且在发电过程中水库水面高度保持不变。发电站将水的势能转化为电能的总效率 $\eta=60 \%$ 。在地球同步轨道上, 太阳光垂直照射时的辐射功率为 $10 P_{0}$ 。太阳能电池板将太阳能转化为电能的效率为 $20 \%$, 将电能输送到地面的过程要损失 50\%。若要使（1）中的宇宙太阳能发电站与三峡电站具有相同的发电能力, 同步卫星上太阳能电池板的面积至少为多大? 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 请记住，你的答案应以$\mathrm{~m}^{2}$为单位计算，但在给出最终答案时，请不要包含单位。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。

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我国计划在 2030 年前实现载人登陆月球开展科学探索, 其后将探索建造月球科研试验站, 开展系统、连续的月球探测和相关技术试验验证。假设质量为 $m$ 的飞船到达月球时, 在距离月面的高度等于月球半径的 $\frac{1}{2}$ 处先绕着月球表面做匀速圆周运动, 其周期为 $T_{1}$, 已知月球的自转周期为 $T_{2}$, 月球的半径为 $R$, 引力常量为 $G$, 下列说法正确的是 ( ) A: 月球的第一宇宙速度为 $\frac{3 \sqrt{3} \pi R}{2 T_{1}}$ B: 月球两极的重力加速度为 $\frac{27 \pi^{2} R}{T_{1}^{2}}$ C: 当飞船停在月球纬度 $60^{\circ}$ 的区域时, 其自转向心加速度为 $\frac{\sqrt{3} \pi^{2} R}{2 T_{2}^{2}}$ D: 当飞船停在月球赤道的水平面上时, 受到的支持力为 $\pi^{2} m R\left(\frac{27}{2 T_{1}^{2}}-\frac{4}{T_{2}^{2}}\right)$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 我国计划在 2030 年前实现载人登陆月球开展科学探索, 其后将探索建造月球科研试验站, 开展系统、连续的月球探测和相关技术试验验证。假设质量为 $m$ 的飞船到达月球时, 在距离月面的高度等于月球半径的 $\frac{1}{2}$ 处先绕着月球表面做匀速圆周运动, 其周期为 $T_{1}$, 已知月球的自转周期为 $T_{2}$, 月球的半径为 $R$, 引力常量为 $G$, 下列说法正确的是 ( ) A: 月球的第一宇宙速度为 $\frac{3 \sqrt{3} \pi R}{2 T_{1}}$ B: 月球两极的重力加速度为 $\frac{27 \pi^{2} R}{T_{1}^{2}}$ C: 当飞船停在月球纬度 $60^{\circ}$ 的区域时, 其自转向心加速度为 $\frac{\sqrt{3} \pi^{2} R}{2 T_{2}^{2}}$ D: 当飞船停在月球赤道的水平面上时, 受到的支持力为 $\pi^{2} m R\left(\frac{27}{2 T_{1}^{2}}-\frac{4}{T_{2}^{2}}\right)$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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中国火星探测器于 2020 年发射, 预计 2021 年到达火星 (火星与太阳的距离大于地球与太阳的距离), 要一次性完成 “环绕、着陆、巡视”三步走。现用 $h$ 表示探测器与火星表面的距离, $a$ 表示探测器所受的火星引力产生的加速度, $a$ 随 $h$ 变化的图像如图所示, 图像中 $a_{1} 、 a_{2} 、 h_{0}$ 为已知, 引力常量为 $G$ 。下列判断正确的是 ( ) [图1] A: 火星绕太阳做圆周运动的线速度小于地球绕太阳做圆周运动的线速度 B: 火星表面的重力加速度大小为 $a_{2}$ C: 火星的半径为 $\frac{\sqrt{a_{1}}}{\sqrt{a_{2}}-\sqrt{a_{1}}} h_{0}$ D: 火星的质量为 $\left(\frac{\sqrt{a_{1} a_{2}}}{\sqrt{a_{2}}-\sqrt{a_{1}}}\right)^{2} \frac{h_{0}^{2}}{2 G}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 中国火星探测器于 2020 年发射, 预计 2021 年到达火星 (火星与太阳的距离大于地球与太阳的距离), 要一次性完成 “环绕、着陆、巡视”三步走。现用 $h$ 表示探测器与火星表面的距离, $a$ 表示探测器所受的火星引力产生的加速度, $a$ 随 $h$ 变化的图像如图所示, 图像中 $a_{1} 、 a_{2} 、 h_{0}$ 为已知, 引力常量为 $G$ 。下列判断正确的是 ( ) [图1] A: 火星绕太阳做圆周运动的线速度小于地球绕太阳做圆周运动的线速度 B: 火星表面的重力加速度大小为 $a_{2}$ C: 火星的半径为 $\frac{\sqrt{a_{1}}}{\sqrt{a_{2}}-\sqrt{a_{1}}} h_{0}$ D: 火星的质量为 $\left(\frac{\sqrt{a_{1} a_{2}}}{\sqrt{a_{2}}-\sqrt{a_{1}}}\right)^{2} \frac{h_{0}^{2}}{2 G}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

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2018 年 12 月 7 日是我国发射“悟空”探测卫星三周年的日子，该卫星的发射为人类对暗物质的研究做出了重大贡献. 假设两颗质量相等的星球绕其球心连线中点转动, 理论计算的周期与实际观测的周期有出入, 且 $\frac{T_{\text {理论 }}}{T_{\text {观测 }}}=\frac{\sqrt{n}}{1}(n>1)$, 科学家推测, 在以两星球球心连线为直径的球体空间中均匀分布着暗物质, 设两星球球心连线长度为 $L$, 质量均为 $m$ ，据此推测, 暗物质的质量为( ) A: $(n-1) m$ B: $n m$ C: $\frac{n-2}{8} m$ D: $\frac{n-1}{4} m$

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: 2018 年 12 月 7 日是我国发射“悟空”探测卫星三周年的日子，该卫星的发射为人类对暗物质的研究做出了重大贡献. 假设两颗质量相等的星球绕其球心连线中点转动, 理论计算的周期与实际观测的周期有出入, 且 $\frac{T_{\text {理论 }}}{T_{\text {观测 }}}=\frac{\sqrt{n}}{1}(n>1)$, 科学家推测, 在以两星球球心连线为直径的球体空间中均匀分布着暗物质, 设两星球球心连线长度为 $L$, 质量均为 $m$ ，据此推测, 暗物质的质量为( ) A: $(n-1) m$ B: $n m$ C: $\frac{n-2}{8} m$ D: $\frac{n-1}{4} m$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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假设在半径为 $R$ 的某天体上发射一颗该天体的卫星, 已知引力常量为 $G$, 忽略该天体自转。 若卫星距该天体表面的高度为 $h$, 测得卫星在该处做圆周运动的周期为 $T_{l}$, 则该天体的密度是多少?

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 假设在半径为 $R$ 的某天体上发射一颗该天体的卫星, 已知引力常量为 $G$, 忽略该天体自转。 若卫星距该天体表面的高度为 $h$, 测得卫星在该处做圆周运动的周期为 $T_{l}$, 则该天体的密度是多少? 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

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Astronomy_619

电影《流浪地球》讲述的是面对太阳快速老化膨胀的灾难, 人类制定了“流浪地球”计划, 这首先需要使自转角速度大小为 $\omega$ 的地球停止自转, 再将地球推移出太阳系到达距离太阳最近的恒星 (比邻星)。为了使地球停止自转, 设想的方案就是在地球赤道上均匀地安装 $N$ 台“喷气”发动机, 如图所示 ( $N$ 较大, 图中只画出了 4 个)。假设每台发动机均能沿赤道的切线方向提供大小恒为 $F$ 的推力, 该推力可阻碍地球的自转。已知描述地球转动的动力学方程与描述质点运动的牛顿第二定律方程 $F=m a$ 具有相似性, 为 $M=I \beta$, 其中 $M$ 为外力的总力矩, 即外力与对应力臂乘积的总和, 其值为 $N F R ; I$ 为地球相对地轴的转动惯量; $\beta$ 为单位时间内地球的角速度的改变量。将地球看成质量分布均匀的球体，下列说法中正确的是（） [图1] A: 在 $M=I \beta$ 与 $F=m a$ 的类比中, 与质量 $m$ 对应的物理量是转动惯量 $I$, 其物理意义是反映改变地球绕地轴转动情况的难易程度 B: 地球自转刹车过程中，赤道表面附近的重力加速度逐渐变小 C: 地球停止自转后, 赤道附近比两极点附近的重力加速度大 D: 地球自转刹车过程中, 两极点的重力加速度逐渐变大 E: 这些行星发动机同时开始工作, 使地球停止自转所需要的时间为 $\frac{\omega I}{N F}$ F: 若发动机“喷气”方向与地球上该点的自转线速度方向相反, 则地球赤道地面的人可能会“飘”起来 G: 在 $M=I \beta$ 与 $F=m a$ 的类比中, 力矩 $M$ 对应的物理量是 $m$, 其物理意义是反映改变地球绕地轴转动情况的难易程度 H: $\beta$ 的单位应为 $\mathrm{rad} / \mathrm{s}$ I: $\beta-t$ 图象中曲线与 $t$ 轴围成的面积的绝对值等于角速度的变化量的大小 J: 地球自转刹车过程中, 赤道表面附近的重力加速度逐渐变大 K: 若停止自转后, 地球仍为均匀球体, 则赤道处附近与极地附近的重力加速度大小没有差异

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: 电影《流浪地球》讲述的是面对太阳快速老化膨胀的灾难, 人类制定了“流浪地球”计划, 这首先需要使自转角速度大小为 $\omega$ 的地球停止自转, 再将地球推移出太阳系到达距离太阳最近的恒星 (比邻星)。为了使地球停止自转, 设想的方案就是在地球赤道上均匀地安装 $N$ 台“喷气”发动机, 如图所示 ( $N$ 较大, 图中只画出了 4 个)。假设每台发动机均能沿赤道的切线方向提供大小恒为 $F$ 的推力, 该推力可阻碍地球的自转。已知描述地球转动的动力学方程与描述质点运动的牛顿第二定律方程 $F=m a$ 具有相似性, 为 $M=I \beta$, 其中 $M$ 为外力的总力矩, 即外力与对应力臂乘积的总和, 其值为 $N F R ; I$ 为地球相对地轴的转动惯量; $\beta$ 为单位时间内地球的角速度的改变量。将地球看成质量分布均匀的球体，下列说法中正确的是（） [图1] A: 在 $M=I \beta$ 与 $F=m a$ 的类比中, 与质量 $m$ 对应的物理量是转动惯量 $I$, 其物理意义是反映改变地球绕地轴转动情况的难易程度 B: 地球自转刹车过程中，赤道表面附近的重力加速度逐渐变小 C: 地球停止自转后, 赤道附近比两极点附近的重力加速度大 D: 地球自转刹车过程中, 两极点的重力加速度逐渐变大 E: 这些行星发动机同时开始工作, 使地球停止自转所需要的时间为 $\frac{\omega I}{N F}$ F: 若发动机“喷气”方向与地球上该点的自转线速度方向相反, 则地球赤道地面的人可能会“飘”起来 G: 在 $M=I \beta$ 与 $F=m a$ 的类比中, 力矩 $M$ 对应的物理量是 $m$, 其物理意义是反映改变地球绕地轴转动情况的难易程度 H: $\beta$ 的单位应为 $\mathrm{rad} / \mathrm{s}$ I: $\beta-t$ 图象中曲线与 $t$ 轴围成的面积的绝对值等于角速度的变化量的大小 J: 地球自转刹车过程中, 赤道表面附近的重力加速度逐渐变大 K: 若停止自转后, 地球仍为均匀球体, 则赤道处附近与极地附近的重力加速度大小没有差异 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, E, F, G, H, I, J, K]

MC

Astronomy

ZH

multi-modal

Astronomy_389

地球和木星绕太阳运行的轨道可以看作是圆形的, 它们各自的卫星轨道也可看作是圆形的. 已知木星的公转轨道半径约为地球公转轨道半径的 5 倍, 木星半径约为地球半径的 11 倍, 木星质量大于地球质量。如图所示是地球和木星的不同卫星做圆周运动的半径 $r$ 的立方与周期 $T$ 的平方的关系图像, 已知万有引力常量为 $G$, 地球的半径为 $R$ 。下列说法正确的是 ( ) [图1] A: 木星与地球的质量之比为 $\frac{b d}{11 a c}$ B: 木星与地球的线速度之比为 $1: 5$ C: 地球密度为 $\frac{3 \pi a}{G d R^{3}}$ D: 木星密度为 $\frac{3 \pi b}{125 G c R^{3}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 地球和木星绕太阳运行的轨道可以看作是圆形的, 它们各自的卫星轨道也可看作是圆形的. 已知木星的公转轨道半径约为地球公转轨道半径的 5 倍, 木星半径约为地球半径的 11 倍, 木星质量大于地球质量。如图所示是地球和木星的不同卫星做圆周运动的半径 $r$ 的立方与周期 $T$ 的平方的关系图像, 已知万有引力常量为 $G$, 地球的半径为 $R$ 。下列说法正确的是 ( ) [图1] A: 木星与地球的质量之比为 $\frac{b d}{11 a c}$ B: 木星与地球的线速度之比为 $1: 5$ C: 地球密度为 $\frac{3 \pi a}{G d R^{3}}$ D: 木星密度为 $\frac{3 \pi b}{125 G c R^{3}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

multi-modal

Astronomy_1016

On $24^{\text {th }}$ August 2016, astronomers discovered a planet orbiting the closest star to the Sun, Proxima Centauri, situated 4.22 light years away, which fulfils a long-standing dream of science-fiction writers: a world that is close enough for humans to send their first interstellar spacecraft. Astronomers have noted how the motion of Proxima Centauri changed in the first months of 2016, with the star moving towards and away from the Earth, as seen in the figure below. Sometimes Proxima Centauri is approaching Earth at $5 \mathrm{~km} \mathrm{hour}^{-1}-$ normal human walking pace - and at times receding at the same speed. This regular pattern of changing radial velocities caused by an unseen planet, which they named Proxima Centauri B, repeats and results in tiny Doppler shifts in the star's light, making the light appear slightly redder, then bluer. [figure1] By considering that the total linear momentum of the star-planet system in the centre of mass frame is zero, estimate the minimum mass of the planet in terms of Earth masses. Why is this a minimum for the mass of the planet?

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: On $24^{\text {th }}$ August 2016, astronomers discovered a planet orbiting the closest star to the Sun, Proxima Centauri, situated 4.22 light years away, which fulfils a long-standing dream of science-fiction writers: a world that is close enough for humans to send their first interstellar spacecraft. Astronomers have noted how the motion of Proxima Centauri changed in the first months of 2016, with the star moving towards and away from the Earth, as seen in the figure below. Sometimes Proxima Centauri is approaching Earth at $5 \mathrm{~km} \mathrm{hour}^{-1}-$ normal human walking pace - and at times receding at the same speed. This regular pattern of changing radial velocities caused by an unseen planet, which they named Proxima Centauri B, repeats and results in tiny Doppler shifts in the star's light, making the light appear slightly redder, then bluer. [figure1] By considering that the total linear momentum of the star-planet system in the centre of mass frame is zero, estimate the minimum mass of the planet in terms of Earth masses. Why is this a minimum for the mass of the planet? 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_797

Order the following phases of the Sun's evolution from first to last chronologically. 13. Helium flash 14. White dwarf 15. Red giant branch 16. Asymptotic giant branch 17. End of hydrogen fusion in the core A: 5, 4, 1, 3, 2 B: $5,3,1,4,2$ C: $1,5,3,4,2$ D: $5,2,4,1,3$ E: $3,5,1,4,2$

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: Order the following phases of the Sun's evolution from first to last chronologically. 13. Helium flash 14. White dwarf 15. Red giant branch 16. Asymptotic giant branch 17. End of hydrogen fusion in the core A: 5, 4, 1, 3, 2 B: $5,3,1,4,2$ C: $1,5,3,4,2$ D: $5,2,4,1,3$ E: $3,5,1,4,2$ You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D, E].

SC

Astronomy

EN

text-only

Astronomy_769

A rocket that has accelerated to the second cosmic velocity can... A: circle around the Earth in a stable orbit. B: circle around the Earth in an elliptic orbit. C: escape the gravitational field of the Earth. D: escape the gravitational field of the Sun.

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: A rocket that has accelerated to the second cosmic velocity can... A: circle around the Earth in a stable orbit. B: circle around the Earth in an elliptic orbit. C: escape the gravitational field of the Earth. D: escape the gravitational field of the Sun. You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

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Astronomy

EN

text-only

Astronomy_85

作为一种新型的多功能航天飞行器, 航天飞机集火箭、卫星和飞机的技术特点于一身。假设一航天飞机在完成某维修任务后, 在 $A$ 点从圆形轨道I进入椭圆轨道II, 如图所示, 已知 $A$ 点距地面的高度为 $2 R$ ( $R$ 为地球半径), $B$ 点为轨道II上的近地点（离地面高度忽略不计), 地表重力加速度为 $g$, 地球质量为 $M$ 。又知若物体在与星球无穷远处时其引力势能为零, 则当物体与星球球心距离为 $r$ 时, 其引力势能 $E_{p}=-\frac{G M m}{r}$ (式中 $m$为物体的质量, $M$ 为星球的质量, $G$ 为引力常量), 不计空气阻力。则下列说法中正确的有 [图1] A: 该航天飞机在轨道 $\mathrm{II}$ 上运动的周期 $T_{2}$ 小于在轨道 $\mathrm{I}$ 上运动的周期 $T_{1}$ B: 该航天飞机在轨道II上经过 $B$ 点的速度大于轨道I上经过 $A$ 点的速度 C: 该航天飞机在轨道II上经过 $B$ 点的速度大于 $7.9 \mathrm{~km} / \mathrm{s}$, 小于 $11.2 \mathrm{~km} / \mathrm{s}$ D: 该航天飞机在轨道II上从 $A$ 运动到 $B$ 的时间为 $\pi \sqrt{\frac{2 R}{g}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 作为一种新型的多功能航天飞行器, 航天飞机集火箭、卫星和飞机的技术特点于一身。假设一航天飞机在完成某维修任务后, 在 $A$ 点从圆形轨道I进入椭圆轨道II, 如图所示, 已知 $A$ 点距地面的高度为 $2 R$ ( $R$ 为地球半径), $B$ 点为轨道II上的近地点（离地面高度忽略不计), 地表重力加速度为 $g$, 地球质量为 $M$ 。又知若物体在与星球无穷远处时其引力势能为零, 则当物体与星球球心距离为 $r$ 时, 其引力势能 $E_{p}=-\frac{G M m}{r}$ (式中 $m$为物体的质量, $M$ 为星球的质量, $G$ 为引力常量), 不计空气阻力。则下列说法中正确的有 [图1] A: 该航天飞机在轨道 $\mathrm{II}$ 上运动的周期 $T_{2}$ 小于在轨道 $\mathrm{I}$ 上运动的周期 $T_{1}$ B: 该航天飞机在轨道II上经过 $B$ 点的速度大于轨道I上经过 $A$ 点的速度 C: 该航天飞机在轨道II上经过 $B$ 点的速度大于 $7.9 \mathrm{~km} / \mathrm{s}$, 小于 $11.2 \mathrm{~km} / \mathrm{s}$ D: 该航天飞机在轨道II上从 $A$ 运动到 $B$ 的时间为 $\pi \sqrt{\frac{2 R}{g}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_993

Light from a star is split into a line spectrum of different colours. The line spectrum from the star is shown, along with the line spectra of some individual elements. Identify the elements present in the star. [figure1] A: Helium and hydrogen B: Potassium and sodium and hydrogen C: Hydrogen and sodium D: Sodium and potassium

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 from a star is split into a line spectrum of different colours. The line spectrum from the star is shown, along with the line spectra of some individual elements. Identify the elements present in the star. [figure1] A: Helium and hydrogen B: Potassium and sodium and hydrogen C: Hydrogen and sodium D: Sodium and potassium 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_255

太阳系的行星几乎在同一平面内沿同一方向绕太阳做圆周运动, 当地球恰好运行到某个行星和太阳之间，且三者几乎成一条直线的现象，天文学成为“行星冲日”据报道， 2014 年各行星冲日时间分别是: 1 月 6 日，木星冲日， 4 月 9 日火星冲日， 6 月 11 日土星冲日, 8 月 29 日, 海王星冲日, 10 月 8 日, 天王星冲日, 已知地球轨道以外的行星绕太阳运动的轨道半径如下表所示，则下列判断正确的是（） | | 地球 | 火星 | 木星 | 土星 | 天王星 | 海王星 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | 轨道半径 (AU) | 1.0 | 1.5 | 5.2 | 9.5 | 19 | 30 | A: 各地外行星每年都会出现冲日现象 B: 在 2015 年内一定会出现木星冲日 C: 天王星相邻两次的冲日的时间是土星的一半 D: 地外行星中海王星相邻两次冲日间隔时间最短

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: 太阳系的行星几乎在同一平面内沿同一方向绕太阳做圆周运动, 当地球恰好运行到某个行星和太阳之间，且三者几乎成一条直线的现象，天文学成为“行星冲日”据报道， 2014 年各行星冲日时间分别是: 1 月 6 日，木星冲日， 4 月 9 日火星冲日， 6 月 11 日土星冲日, 8 月 29 日, 海王星冲日, 10 月 8 日, 天王星冲日, 已知地球轨道以外的行星绕太阳运动的轨道半径如下表所示，则下列判断正确的是（） | | 地球 | 火星 | 木星 | 土星 | 天王星 | 海王星 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | 轨道半径 (AU) | 1.0 | 1.5 | 5.2 | 9.5 | 19 | 30 | A: 各地外行星每年都会出现冲日现象 B: 在 2015 年内一定会出现木星冲日 C: 天王星相邻两次的冲日的时间是土星的一半 D: 地外行星中海王星相邻两次冲日间隔时间最短 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_902

An observer on the equator sees the Moon rise at 22:00. Ignoring the inclination and eccentricity of the Moon's orbit, when will it rise the next night? The Moon's orbital period is 27.3 days. A: 21:07 B: $21: 47$ C: $22: 13$ D: $22: 53$

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: An observer on the equator sees the Moon rise at 22:00. Ignoring the inclination and eccentricity of the Moon's orbit, when will it rise the next night? The Moon's orbital period is 27.3 days. A: 21:07 B: $21: 47$ C: $22: 13$ D: $22: 53$ 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_126

在发射卫星时, 往往先将卫星发送到一个粗圆轨道上, 再变轨到圆轨道。已知某卫星运行的椭圆轨道的近地点 $M$ 距地面 $210 \mathrm{~km}$, 远地点 $N$ 距地面 $345 \mathrm{~km}$, 卫星进入该轨道正常运行时, 通过 $M$ 点和 $N$ 点时的速率分别为 $v_{1}$ 和 $v_{2}$, 当某次卫星通过 $N$ 点时, 启动卫星上的发动机, 使卫星在短时间内加速后进入离地面 $345 \mathrm{~km}$ 的圆形轨道, 开始绕地球做匀速圆周运动, 这时卫星的速率为 $v_{3}$ 。比较卫星在 $M 、 N 、 P$ 三点正常运行时（不包括启动发动机加速阶段）的速率 $v_{1} 、 v_{2} 、 v_{3}$ 和加速度大小 $a_{1} 、 a_{2} 、 a_{3}$, 下列结论正确的是 ( ) [图1] A: $v_{1}>v_{2}>v_{3}, a_{1}>a_{2}=a_{3}$ B: $v_{1}>v_{2}=v_{3}, a_{1}>a_{2}>a_{3}$ C: $v_{1}>v_{3}>v_{2}, a_{1}>a_{3}>a_{2}$ D: $v_{1}>v_{3}>v_{2}, a_{1}>a_{2}=a_{3}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 在发射卫星时, 往往先将卫星发送到一个粗圆轨道上, 再变轨到圆轨道。已知某卫星运行的椭圆轨道的近地点 $M$ 距地面 $210 \mathrm{~km}$, 远地点 $N$ 距地面 $345 \mathrm{~km}$, 卫星进入该轨道正常运行时, 通过 $M$ 点和 $N$ 点时的速率分别为 $v_{1}$ 和 $v_{2}$, 当某次卫星通过 $N$ 点时, 启动卫星上的发动机, 使卫星在短时间内加速后进入离地面 $345 \mathrm{~km}$ 的圆形轨道, 开始绕地球做匀速圆周运动, 这时卫星的速率为 $v_{3}$ 。比较卫星在 $M 、 N 、 P$ 三点正常运行时（不包括启动发动机加速阶段）的速率 $v_{1} 、 v_{2} 、 v_{3}$ 和加速度大小 $a_{1} 、 a_{2} 、 a_{3}$, 下列结论正确的是 ( ) [图1] A: $v_{1}>v_{2}>v_{3}, a_{1}>a_{2}=a_{3}$ B: $v_{1}>v_{2}=v_{3}, a_{1}>a_{2}>a_{3}$ C: $v_{1}>v_{3}>v_{2}, a_{1}>a_{3}>a_{2}$ D: $v_{1}>v_{3}>v_{2}, a_{1}>a_{2}=a_{3}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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某行星的卫星 $\mathrm{A} 、 \mathrm{~B}$ 绕以其为焦点的椭圆轨道运行, 作用于 $\mathrm{A} 、 \mathrm{~B}$ 的引力随时间的变化如图所示, 其中 $t_{2}=\sqrt{2} t_{1}$, 行星到卫星 $\mathrm{A} 、 \mathrm{~B}$ 轨道上点的距离分别记为 $r_{A} 、 r_{B}$ 。假设 $\mathrm{A} 、 \mathrm{~B}$ 只受到行星的引力, 下列叙述正确的是 ( ) [图1] A: B 与 A 的绕行周期之比为 $\sqrt{2}: 1$ B: $r_{B}$ 的最大值与 $r_{B}$ 的最小值之比为 $2: 1$ C: $r_{A}$ 的最大值与 $r_{A}$ 的最小值之比为 3:1 D: $r_{B}$ 的最小值小于 $r_{A}$ 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。 这是一个单选题（只有一个正确答案）。 问题: 某行星的卫星 $\mathrm{A} 、 \mathrm{~B}$ 绕以其为焦点的椭圆轨道运行, 作用于 $\mathrm{A} 、 \mathrm{~B}$ 的引力随时间的变化如图所示, 其中 $t_{2}=\sqrt{2} t_{1}$, 行星到卫星 $\mathrm{A} 、 \mathrm{~B}$ 轨道上点的距离分别记为 $r_{A} 、 r_{B}$ 。假设 $\mathrm{A} 、 \mathrm{~B}$ 只受到行星的引力, 下列叙述正确的是 ( ) [图1] A: B 与 A 的绕行周期之比为 $\sqrt{2}: 1$ B: $r_{B}$ 的最大值与 $r_{B}$ 的最小值之比为 $2: 1$ C: $r_{A}$ 的最大值与 $r_{A}$ 的最小值之比为 3:1 D: $r_{B}$ 的最小值小于 $r_{A}$ 的最大值 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy_585

2019 年 4 月 11 日 21 时黑洞视界望远镜合作组织(ETE)宣布了近邻巨粗圆星系 M87 中心捕获的首张黑洞图像, 提供了黑洞存在的直接“视觉”证据, 验证了 1915 年爱因斯坦的伟大预言。一种理论认为，整个宇宙很可能是个黑洞，如今可观测宇宙的范围膨胀到了半径 465 亿光年的规模, 也就是说, 我们的宇宙就像一个直径 930 亿光年的球体。黑洞的质量 $M$ 和半径 $R$ 的关系满足史瓦西半径公式 $\frac{M}{R}=\frac{c^{2}}{2 G}$ (其中 $\mathrm{c}$ 为光速, 其值为 $c=3 \times 10^{8} \mathrm{~m} / \mathrm{s}, G$ 为引力常量, 其值为 $6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2}$ ) 则, 由此可估算出宇宙的总质量的数量级约为 ( ) A: $10^{54} \mathrm{~kg}$ B: $10^{44} \mathrm{~kg}$ C: $10^{34} \mathrm{~kg}$ D: $10^{24} \mathrm{~kg}$

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: 2019 年 4 月 11 日 21 时黑洞视界望远镜合作组织(ETE)宣布了近邻巨粗圆星系 M87 中心捕获的首张黑洞图像, 提供了黑洞存在的直接“视觉”证据, 验证了 1915 年爱因斯坦的伟大预言。一种理论认为，整个宇宙很可能是个黑洞，如今可观测宇宙的范围膨胀到了半径 465 亿光年的规模, 也就是说, 我们的宇宙就像一个直径 930 亿光年的球体。黑洞的质量 $M$ 和半径 $R$ 的关系满足史瓦西半径公式 $\frac{M}{R}=\frac{c^{2}}{2 G}$ (其中 $\mathrm{c}$ 为光速, 其值为 $c=3 \times 10^{8} \mathrm{~m} / \mathrm{s}, G$ 为引力常量, 其值为 $6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2}$ ) 则, 由此可估算出宇宙的总质量的数量级约为 ( ) A: $10^{54} \mathrm{~kg}$ B: $10^{44} \mathrm{~kg}$ C: $10^{34} \mathrm{~kg}$ D: $10^{24} \mathrm{~kg}$ 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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“势阱” 是量子力学中的常见概念, 在经典力学中也有体现。当粒子在某力场中运动,其势能函数曲线在空间某范围内存在最小值，形如陷阶，粒子很难跑出来。各种形式的势能函数只要具有这种特点, 我们都可以称它为势阱, 比如重力势阱、引力势阱、弹力势阱等。 如图甲所示, 光滑轨道 $a b c$ 固定在坚直平面内形成一重力势阱, 两侧高分别为 $k H$ 和 $H$ 。一可视为质点的质量为 $m$ 的小球, 静置于水平轨道 $b$ 处。已知重力加速度为 $g$; 以 $a$ 处所在平面为重力势能面, 写出该小球在 $b$ 处机械能的表达式; [图1] 甲 [图2] 乙 [图3] 丙

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: “势阱” 是量子力学中的常见概念, 在经典力学中也有体现。当粒子在某力场中运动,其势能函数曲线在空间某范围内存在最小值，形如陷阶，粒子很难跑出来。各种形式的势能函数只要具有这种特点, 我们都可以称它为势阱, 比如重力势阱、引力势阱、弹力势阱等。 如图甲所示, 光滑轨道 $a b c$ 固定在坚直平面内形成一重力势阱, 两侧高分别为 $k H$ 和 $H$ 。一可视为质点的质量为 $m$ 的小球, 静置于水平轨道 $b$ 处。已知重力加速度为 $g$; 以 $a$ 处所在平面为重力势能面, 写出该小球在 $b$ 处机械能的表达式; [图1] 甲 [图2] 乙 [图3] 丙 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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2014 年 12 月 14 日, 北京飞行控制中心传来好消息, 嫦娥三号探测器平稳落月, 已知嫦娥三号探测器在地球表面受的重力为 $G_{1}$, 绕月球表面飞行时受到月球的引力为 $G_{2}$,地球的半径为 $R_{1}$, 月球的半径为 $R_{2}$, 地球表面处的重力加速为 $g$, 则 ( ) A: 月球与地球的质量之比为 $\frac{G_{1} R_{2}{ }^{2}}{G_{2} R_{1}{ }^{2}}$ B: 月球表面处的重力加速度为 $\frac{G_{1}}{G_{2}} g$ C: 月球与地球的第一宇宙速度之比为 $\sqrt{\frac{R_{2} G_{1}}{R_{1} G_{2}}}$ D: 探测器沿月球表面轨道上做匀速圆周运动的周期为 $2 \pi \sqrt{\frac{R_{2} G_{1}}{g G_{2}}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 2014 年 12 月 14 日, 北京飞行控制中心传来好消息, 嫦娥三号探测器平稳落月, 已知嫦娥三号探测器在地球表面受的重力为 $G_{1}$, 绕月球表面飞行时受到月球的引力为 $G_{2}$,地球的半径为 $R_{1}$, 月球的半径为 $R_{2}$, 地球表面处的重力加速为 $g$, 则 ( ) A: 月球与地球的质量之比为 $\frac{G_{1} R_{2}{ }^{2}}{G_{2} R_{1}{ }^{2}}$ B: 月球表面处的重力加速度为 $\frac{G_{1}}{G_{2}} g$ C: 月球与地球的第一宇宙速度之比为 $\sqrt{\frac{R_{2} G_{1}}{R_{1} G_{2}}}$ D: 探测器沿月球表面轨道上做匀速圆周运动的周期为 $2 \pi \sqrt{\frac{R_{2} G_{1}}{g G_{2}}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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北京时间 2019 年 4 月 10 日，人类历史上首张黑洞“照片”（如图）被正式披露，引起世界轰动; 2020 年 4 月 7 日“事件视界望远镜（EHT）”项目组公布了第二张黑洞“照片”，呈现了更多有关黑洞的信息。黑洞是质量极大的天体，引力极强。一个事件刚好能被观察到的那个时空界面称为视界。例如, 发生在黑洞里的事件不会被黑洞外的人所观察到，因此我们可以把黑洞的视界作为黑洞的“边界”。在黑洞视界范围内，连光也不能逃逸。由于黑洞质量极大，其周围时空严重变形。这样，即使是被黑洞挡着的恒星发出的光, 有一部分光会落入黑洞中, 但还有另一部分离黑洞较远的光线会绕过黑洞, 通过弯曲的路径到达地球。根据上述材料, 结合所学知识判断下列说法正确的是 [图1] A: 黑洞“照片”明亮部分是地球上的观测者捕捉到的黑洞自身所发出的光 B: 地球观测者看到的黑洞“正后方”的几个恒星之间的距离比实际的远 C: 视界是真实的物质面, 只是外部观测者对它一无所知 D: 黑洞的第二宇宙速度小于光速 $c$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 北京时间 2019 年 4 月 10 日，人类历史上首张黑洞“照片”（如图）被正式披露，引起世界轰动; 2020 年 4 月 7 日“事件视界望远镜（EHT）”项目组公布了第二张黑洞“照片”，呈现了更多有关黑洞的信息。黑洞是质量极大的天体，引力极强。一个事件刚好能被观察到的那个时空界面称为视界。例如, 发生在黑洞里的事件不会被黑洞外的人所观察到，因此我们可以把黑洞的视界作为黑洞的“边界”。在黑洞视界范围内，连光也不能逃逸。由于黑洞质量极大，其周围时空严重变形。这样，即使是被黑洞挡着的恒星发出的光, 有一部分光会落入黑洞中, 但还有另一部分离黑洞较远的光线会绕过黑洞, 通过弯曲的路径到达地球。根据上述材料, 结合所学知识判断下列说法正确的是 [图1] A: 黑洞“照片”明亮部分是地球上的观测者捕捉到的黑洞自身所发出的光 B: 地球观测者看到的黑洞“正后方”的几个恒星之间的距离比实际的远 C: 视界是真实的物质面, 只是外部观测者对它一无所知 D: 黑洞的第二宇宙速度小于光速 $c$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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如图所示, 地球和行星绕太阳做匀速圆周运动, 且此时行星、地球、太阳三者共线 $\cdot$地球和行星做匀速圆周运动的半径 $r_{1} 、 r_{2}$ 之比为 $r_{1}: r_{2}=1: 4$, 不计地球和行星之间的相互影响$\cdot$下列说法正确的是 [图1] A: 行星绕太阳做圆周运动的周期为 8 年 B: 地球和行星的线速度大小之比为 $1: 2$ C: 至少经过 $\frac{8}{7}$ 年, 地球位于太阳和行星连线之间 D: 经过相同时间, 地球和行星半径扫过的面积之比为 1:2

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 如图所示, 地球和行星绕太阳做匀速圆周运动, 且此时行星、地球、太阳三者共线 $\cdot$地球和行星做匀速圆周运动的半径 $r_{1} 、 r_{2}$ 之比为 $r_{1}: r_{2}=1: 4$, 不计地球和行星之间的相互影响$\cdot$下列说法正确的是 [图1] A: 行星绕太阳做圆周运动的周期为 8 年 B: 地球和行星的线速度大小之比为 $1: 2$ C: 至少经过 $\frac{8}{7}$ 年, 地球位于太阳和行星连线之间 D: 经过相同时间, 地球和行星半径扫过的面积之比为 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应为两个或更多的选项：[A, B, C, D]

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脉冲星的本质是中子星, 具有在地面实验室无法实现的极端物理性质, 是理想的天体物理实验室, 对其进行研究, 有希望得到许多重大物理学问题的答案。譬如: 脉冲星的自转周期极棒稳定, 准确的时钟信号为强力波探测。航天器导航等重大科学及技术应用提供了理想工具。2017 年 8 月我国 FAST 天文望远镜首次发现两颗太空脉冲星, 其 中一颗的自转周期为 $\mathrm{T}$ (实际测量为 $1.83 \mathrm{~s}$, 距离地球 1.6 万光年), 假设该星球恰好能维持自转而不瓦解; 地球可视为球体, 其自转周期为 $\mathrm{T}_{0}$, 同一物体在地球赤道上用弹簧科测得的重力为两极处的 0.9 倍, 已知万有引力常量为 $\mathrm{G}$, 则该脉冲星的平均密度 $\rho$及其与地球的平均密度 $\rho_{0}$ 之比正确的是 ( ) A: $\rho=\frac{3 \pi}{G T^{2}}$ B: $\frac{\rho}{\rho_{0}}=\frac{T_{0}^{2}}{10 T^{2}}$ C: $\rho_{0}=\frac{3 \pi}{G T_{0}^{2}}$ D: $\frac{\rho}{\rho_{0}}=\frac{10 T_{0}^{2}}{T^{2}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 脉冲星的本质是中子星, 具有在地面实验室无法实现的极端物理性质, 是理想的天体物理实验室, 对其进行研究, 有希望得到许多重大物理学问题的答案。譬如: 脉冲星的自转周期极棒稳定, 准确的时钟信号为强力波探测。航天器导航等重大科学及技术应用提供了理想工具。2017 年 8 月我国 FAST 天文望远镜首次发现两颗太空脉冲星, 其 中一颗的自转周期为 $\mathrm{T}$ (实际测量为 $1.83 \mathrm{~s}$, 距离地球 1.6 万光年), 假设该星球恰好能维持自转而不瓦解; 地球可视为球体, 其自转周期为 $\mathrm{T}_{0}$, 同一物体在地球赤道上用弹簧科测得的重力为两极处的 0.9 倍, 已知万有引力常量为 $\mathrm{G}$, 则该脉冲星的平均密度 $\rho$及其与地球的平均密度 $\rho_{0}$ 之比正确的是 ( ) A: $\rho=\frac{3 \pi}{G T^{2}}$ B: $\frac{\rho}{\rho_{0}}=\frac{T_{0}^{2}}{10 T^{2}}$ C: $\rho_{0}=\frac{3 \pi}{G T_{0}^{2}}$ D: $\frac{\rho}{\rho_{0}}=\frac{10 T_{0}^{2}}{T^{2}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

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Astronomy_423

万有引力定律 $F_{\text {引 }}=G \frac{m_{1} m_{2}}{r^{2}}$ 和库仑定律 $F_{\text {电 }}=k \frac{q_{1} q_{2}}{r^{2}}$ 都满足力与距离平方成反比关系。如图所示, 计算物体从距离地球球心 $r_{1}$ 处, 远离至与地心距离 $r_{2}$ 处, 万有引力对物体做功时, 由于力的大小随距离而变化, 一般需采用微元法。也可采用从 $r_{1}$ 到 $r_{2}$ 过程的平均力, 即 $\overline{F_{\text {引 }}}=G \frac{m_{1} m_{2}}{r_{1} \cdot r_{2}}$ 计算做功。已知物体质量为 $m$, 地球质量为 $M$, 半径为 $R$, 引力常量为 $G$ 。 求该物体从距离地心 $r_{1}$ 处至距离地心 $r_{2}$ 处的过程中, 万有引力对物体做功 $W$; [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: 万有引力定律 $F_{\text {引 }}=G \frac{m_{1} m_{2}}{r^{2}}$ 和库仑定律 $F_{\text {电 }}=k \frac{q_{1} q_{2}}{r^{2}}$ 都满足力与距离平方成反比关系。如图所示, 计算物体从距离地球球心 $r_{1}$ 处, 远离至与地心距离 $r_{2}$ 处, 万有引力对物体做功时, 由于力的大小随距离而变化, 一般需采用微元法。也可采用从 $r_{1}$ 到 $r_{2}$ 过程的平均力, 即 $\overline{F_{\text {引 }}}=G \frac{m_{1} m_{2}}{r_{1} \cdot r_{2}}$ 计算做功。已知物体质量为 $m$, 地球质量为 $M$, 半径为 $R$, 引力常量为 $G$ 。 求该物体从距离地心 $r_{1}$ 处至距离地心 $r_{2}$ 处的过程中, 万有引力对物体做功 $W$; [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

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multi-modal

Astronomy_862

How far from the Solar System would a galaxy with a redshift of 0.035 be? A: $150 \mathrm{Mpc}$ B: $200 \mathrm{Mpc}$ C: $250 \mathrm{Mpc}$ D: $300 \mathrm{Mpc}$ E: $350 \mathrm{Mpc}$

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 from the Solar System would a galaxy with a redshift of 0.035 be? A: $150 \mathrm{Mpc}$ B: $200 \mathrm{Mpc}$ C: $250 \mathrm{Mpc}$ D: $300 \mathrm{Mpc}$ E: $350 \mathrm{Mpc}$ 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_192

有人设想: 如果在地球的赤道上坚直向上建一座非常高的高楼, 是否可以在楼上直接释放人造卫星呢? 高楼设想图如图所示。已知地球自转周期为 $T$, 同步卫星轨道半径为 $r$; 高楼上有 $A 、 B$ 两个可视为点的小房间, $A$ 到地心的距离为 $\frac{r}{2}, B$ 到地心的距离为 $\frac{3 r}{2}$ 。从 $B$ 房间窗口发射一颗小卫星, 要让这颗小卫星能在与 $B$ 点等高的圆轨道上绕地心与地球自转同向运行, 应给这颗卫星加速还是减速？ $\Delta v$ 大小多少？[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: 有人设想: 如果在地球的赤道上坚直向上建一座非常高的高楼, 是否可以在楼上直接释放人造卫星呢? 高楼设想图如图所示。已知地球自转周期为 $T$, 同步卫星轨道半径为 $r$; 高楼上有 $A 、 B$ 两个可视为点的小房间, $A$ 到地心的距离为 $\frac{r}{2}, B$ 到地心的距离为 $\frac{3 r}{2}$ 。从 $B$ 房间窗口发射一颗小卫星, 要让这颗小卫星能在与 $B$ 点等高的圆轨道上绕地心与地球自转同向运行, 应给这颗卫星加速还是减速？ $\Delta v$ 大小多少？[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：[加速还是减速, 速度变化大小] 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

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Astronomy

ZH

multi-modal

Astronomy_989

Which of these constellations is entirely south of the ecliptic? A: Orion B: Andromeda C: Aquila D: Ophiuchus

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 these constellations is entirely south of the ecliptic? A: Orion B: Andromeda C: Aquila D: Ophiuchus 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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text-only

Astronomy_465

在星球 $\mathrm{M}$ 上将一轻弹簧坚直固定在水平桌面上, 把物体 $\mathrm{P}$ 轻放在弹簧上端, $\mathrm{P}$ 由静止向下运动, 物体的加速度 $a$ 与弹簧的压缩量 $x$ 间的关系如图中实线所示。在另一星球 $\mathrm{N}$ 上用完全相同的弹簧, 改用物体 $\mathrm{Q}$ 完成同样的过程, 其 $a-x$ 关系如图中虚线所示,假设两星球均为质量均匀分布的球体。已知星球 $\mathrm{M}$ 的半径是星球 $\mathrm{N}$ 的 3 倍, 求: 星球 $\mathrm{M}$ 和星球 $\mathrm{N}$ 的密度之比为多少; [图1]

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个数值。 问题: 在星球 $\mathrm{M}$ 上将一轻弹簧坚直固定在水平桌面上, 把物体 $\mathrm{P}$ 轻放在弹簧上端, $\mathrm{P}$ 由静止向下运动, 物体的加速度 $a$ 与弹簧的压缩量 $x$ 间的关系如图中实线所示。在另一星球 $\mathrm{N}$ 上用完全相同的弹簧, 改用物体 $\mathrm{Q}$ 完成同样的过程, 其 $a-x$ 关系如图中虚线所示,假设两星球均为质量均匀分布的球体。已知星球 $\mathrm{M}$ 的半径是星球 $\mathrm{N}$ 的 3 倍, 求: 星球 $\mathrm{M}$ 和星球 $\mathrm{N}$ 的密度之比为多少; [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是数值。

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Astronomy

ZH

multi-modal

Astronomy_1012

The Universe consists of three main components: radiation (including neutrinos), matter (both atoms and dark matter), and dark energy. The overall density of the universe has been dominated by the density of each of those in turn at different times in its history, leading to three different epochs (shown in Fig 3). [figure1] Figure 3: A outline of the three main epochs in the history of the Universe. You do not need to read any data off this graph to answer this question. Credit: Pearson Education, Inc. The scale factor, $a$, describes how the Universe has expanded (i.e. a measure of the relative radius of the Universe), and the current value is defined as $a_{0} \equiv 1$ where the subscript ' 0 ' indicates it is as measured today. At earlier times $a<1$ and at the Big Bang $a=0$. The redshift of an object, $z$, is related to the scale factor as $a=(1+z)^{-1}$ and so the redshift corresponding to now is $z=0$, and far away objects have higher redshift $(z>0)$ since we observe them as they were long ago when the scale factor was smaller. For a Universe to be flat (i.e. zero curvature), its average density must be equal to the critical density, $$ \rho_{\text {crit }, 0}=\frac{3 H_{0}^{2}}{8 \pi G}, $$ where $H_{0}$ is the Hubble constant, measured in 2018 from the cosmic microwave background by the Planck spacecraft to be $67.36 \mathrm{~km} \mathrm{~s}^{-1} \mathrm{Mpc}^{-1}$. The density of the $i^{\text {th }}$ component of the Universe can be expressed relative to the critical density as the density parameter, $$ \Omega_{i}=\frac{\rho_{i}}{\rho_{\text {crit }}} . $$ Planck measured the current density parameters of dark energy and matter as $\Omega_{\Lambda, 0}=0.6847$ and $\Omega_{m, 0}=0.3153$ respectively. In each epoch, the scale factor increases at a different rate with time, $t$, as the density also varies differently with scale factor. - Radiation-dominated epoch: The Universe's early history, where $\rho \propto a^{-4}$ and so $a \propto t^{1 / 2}$ - Matter-dominated epoch: This represents much of the history of the Universe, where $\rho \propto$ $a^{-3}$ and so $a \propto t^{2 / 3}$ - Dark-energy-dominated epoch: This is an era we have recently entered and will remain in for the rest of time, where $\rho$ doesn't vary with scale factor (i.e. is a constant) and so $a \propto e^{H_{0} t}$ Find the time, $t_{D E}$, when the current epoch began (i.e. when the densities of dark energy and matter were equal), given that the age of the Universe today is $t_{0}=13.80 \mathrm{Gyr}$ (where 1 Gyr $=10^{9}$ years). Give you answer in Gyr. You do not need to read anything off the graph.

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 Universe consists of three main components: radiation (including neutrinos), matter (both atoms and dark matter), and dark energy. The overall density of the universe has been dominated by the density of each of those in turn at different times in its history, leading to three different epochs (shown in Fig 3). [figure1] Figure 3: A outline of the three main epochs in the history of the Universe. You do not need to read any data off this graph to answer this question. Credit: Pearson Education, Inc. The scale factor, $a$, describes how the Universe has expanded (i.e. a measure of the relative radius of the Universe), and the current value is defined as $a_{0} \equiv 1$ where the subscript ' 0 ' indicates it is as measured today. At earlier times $a<1$ and at the Big Bang $a=0$. The redshift of an object, $z$, is related to the scale factor as $a=(1+z)^{-1}$ and so the redshift corresponding to now is $z=0$, and far away objects have higher redshift $(z>0)$ since we observe them as they were long ago when the scale factor was smaller. For a Universe to be flat (i.e. zero curvature), its average density must be equal to the critical density, $$ \rho_{\text {crit }, 0}=\frac{3 H_{0}^{2}}{8 \pi G}, $$ where $H_{0}$ is the Hubble constant, measured in 2018 from the cosmic microwave background by the Planck spacecraft to be $67.36 \mathrm{~km} \mathrm{~s}^{-1} \mathrm{Mpc}^{-1}$. The density of the $i^{\text {th }}$ component of the Universe can be expressed relative to the critical density as the density parameter, $$ \Omega_{i}=\frac{\rho_{i}}{\rho_{\text {crit }}} . $$ Planck measured the current density parameters of dark energy and matter as $\Omega_{\Lambda, 0}=0.6847$ and $\Omega_{m, 0}=0.3153$ respectively. In each epoch, the scale factor increases at a different rate with time, $t$, as the density also varies differently with scale factor. - Radiation-dominated epoch: The Universe's early history, where $\rho \propto a^{-4}$ and so $a \propto t^{1 / 2}$ - Matter-dominated epoch: This represents much of the history of the Universe, where $\rho \propto$ $a^{-3}$ and so $a \propto t^{2 / 3}$ - Dark-energy-dominated epoch: This is an era we have recently entered and will remain in for the rest of time, where $\rho$ doesn't vary with scale factor (i.e. is a constant) and so $a \propto e^{H_{0} t}$ Find the time, $t_{D E}$, when the current epoch began (i.e. when the densities of dark energy and matter were equal), given that the age of the Universe today is $t_{0}=13.80 \mathrm{Gyr}$ (where 1 Gyr $=10^{9}$ years). Give you answer in Gyr. You do not need to read anything off the graph. 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 Gyr, 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_458

太阳系各行星几乎在同一平面内沿同一方向绕太阳做圆周运动。当地球恰好运行到某地外行星和太阳之间，且三者几乎排成一条直线的现象，天文学称为“行星冲日”。已知地球及各地外行星绕太阳运动的轨道半径如下表所示, 天文单位用符号 AU 表示。则 | 行星 | 地球 | 火星 | 木星 | 土星 | 天王星 | 海王星 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | 轨道半径 $r / \mathrm{AU}$ | 1.0 | 1.5 | 5.2 | 9.5 | 19 | 30 | A: 木星相邻两次冲日的时间间隔约为 1.1 年 B: 木星的环绕周期约为 25 年 C: 天王星的环绕速度约为土星的两倍 D: 地外行星中, 海王星相邻两次冲日的时间间隔最长

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 太阳系各行星几乎在同一平面内沿同一方向绕太阳做圆周运动。当地球恰好运行到某地外行星和太阳之间，且三者几乎排成一条直线的现象，天文学称为“行星冲日”。已知地球及各地外行星绕太阳运动的轨道半径如下表所示, 天文单位用符号 AU 表示。则 | 行星 | 地球 | 火星 | 木星 | 土星 | 天王星 | 海王星 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | 轨道半径 $r / \mathrm{AU}$ | 1.0 | 1.5 | 5.2 | 9.5 | 19 | 30 | A: 木星相邻两次冲日的时间间隔约为 1.1 年 B: 木星的环绕周期约为 25 年 C: 天王星的环绕速度约为土星的两倍 D: 地外行星中, 海王星相邻两次冲日的时间间隔最长 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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text-only

Astronomy_906

What is the declination of the Sun on the Winter solstice? A: $+45^{\circ}$ B: $+23.5^{\circ}$ C: $0^{\circ}$ D: $-23.5^{\circ}$

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: What is the declination of the Sun on the Winter solstice? A: $+45^{\circ}$ B: $+23.5^{\circ}$ C: $0^{\circ}$ D: $-23.5^{\circ}$ 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].

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Astronomy

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text-only

Astronomy_536

中国北斗卫星导航系统 (BDS) 是中国自行研制的全球卫星导航系统, 是继美国全球定位系统（GPS）、俄罗斯格洛纳斯卫星导航系统（GLONASS）之后第三个成熟的卫星导航系统。2020 年北斗卫星导航系统已形成全球覆盖能力。如图所示是北斗导航系统中部分卫星的轨道示意图, 已知 $a 、 b 、 c$ 三颗卫星均做匀速圆周运动, $a$ 是地球同步卫星，则（） [图1] A: 卫星 $a$ 的运行速度大于卫星 $c$ 的运行速度 B: 卫星 $c$ 的加速度大于卫星 $b$ 的加速度 C: 卫星 $c$ 的运行速度小于第一宇宙速度 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。 这是一个多选题（有多个正确答案）。 问题: 中国北斗卫星导航系统 (BDS) 是中国自行研制的全球卫星导航系统, 是继美国全球定位系统（GPS）、俄罗斯格洛纳斯卫星导航系统（GLONASS）之后第三个成熟的卫星导航系统。2020 年北斗卫星导航系统已形成全球覆盖能力。如图所示是北斗导航系统中部分卫星的轨道示意图, 已知 $a 、 b 、 c$ 三颗卫星均做匀速圆周运动, $a$ 是地球同步卫星，则（） [图1] A: 卫星 $a$ 的运行速度大于卫星 $c$ 的运行速度 B: 卫星 $c$ 的加速度大于卫星 $b$ 的加速度 C: 卫星 $c$ 的运行速度小于第一宇宙速度 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应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

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multi-modal

Astronomy_662

如图所示。甲、乙为地球赤道面内围绕地球运转的通讯卫星。已知甲是与地面相对静止的同步卫星; 乙的运转方向与地球自转方向相反, 轨道半径为地球半径的 2 倍, 周期为 $T$, 在地球赤道上的 $P$ 点有一位观测者，观测者始终相对于地面静止。若地球半径为 $R$, 地球的自转周期为 $T_{0}$ 。求: 若甲、乙之间可进行无线信号通讯, 不计信号传输时间, 甲卫星对地球的最大视角为 $\theta$ ，则甲、乙卫星间信号连续中断的最长时间是多少? [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。甲、乙为地球赤道面内围绕地球运转的通讯卫星。已知甲是与地面相对静止的同步卫星; 乙的运转方向与地球自转方向相反, 轨道半径为地球半径的 2 倍, 周期为 $T$, 在地球赤道上的 $P$ 点有一位观测者，观测者始终相对于地面静止。若地球半径为 $R$, 地球的自转周期为 $T_{0}$ 。求: 若甲、乙之间可进行无线信号通讯, 不计信号传输时间, 甲卫星对地球的最大视角为 $\theta$ ，则甲、乙卫星间信号连续中断的最长时间是多少? [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

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multi-modal

Astronomy_259

2016 年 10 月 17 日 7 时 30 分, 中国在酒泉卫星发射中心使用长征二号 FY11 运载火箭将神舟十一号载人飞船送入太空, 2016 年 10 月 19 日凌晨, 神舟十一号飞船与天宫二号自动交会对接成功, 过去神舟十号与天宫一号对接时, 轨道高度是 343 公里, 而神舟十一号和天宫二号对接时的轨道高度是 393 公里, 比过去高了 50 公里. 由以上信息下列说法正确的是（ ） A: 天宫一号的运行速度小于天宫二号的运行速度 B: 天宫一号的运行周期小于天宫二号的运行周期 C: 神舟十一号飞船如果从 343 公里的轨道变轨到 393 公里的对接轨道机械能减小 D: 天宫一号的加速度小于天宫二号的的加速度

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: 2016 年 10 月 17 日 7 时 30 分, 中国在酒泉卫星发射中心使用长征二号 FY11 运载火箭将神舟十一号载人飞船送入太空, 2016 年 10 月 19 日凌晨, 神舟十一号飞船与天宫二号自动交会对接成功, 过去神舟十号与天宫一号对接时, 轨道高度是 343 公里, 而神舟十一号和天宫二号对接时的轨道高度是 393 公里, 比过去高了 50 公里. 由以上信息下列说法正确的是（ ） A: 天宫一号的运行速度小于天宫二号的运行速度 B: 天宫一号的运行周期小于天宫二号的运行周期 C: 神舟十一号飞船如果从 343 公里的轨道变轨到 393 公里的对接轨道机械能减小 D: 天宫一号的加速度小于天宫二号的的加速度 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]

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Astronomy

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Astronomy_149

已知地球半径为 $R$, 地球表面的重力加速度为 $g$ 。质量为 $m$ 的宇宙飞船在半径为 $2 R$的轨道 1 上绕地球中心 $O$ 做圆两运动。现飞船在轨道 1 的 $A$ 点加速到陏圆轨道 2 上,再在远地点 $B$ 点加速, 从而使飞船转移到半径为 $4 R$ 的轨道 3 上, 如图所示。若相距 $r$的两物体间引力势能为 $E_{\mathrm{p}}=-G \frac{M m}{r}$, 求: 飞船在轨道 2 上经过近地点 $A$ 和远地点 $B$ 的速率之比 $\left(v_{A}: v_{B}\right)$ 。 [图1]

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个数值。 问题: 已知地球半径为 $R$, 地球表面的重力加速度为 $g$ 。质量为 $m$ 的宇宙飞船在半径为 $2 R$的轨道 1 上绕地球中心 $O$ 做圆两运动。现飞船在轨道 1 的 $A$ 点加速到陏圆轨道 2 上,再在远地点 $B$ 点加速, 从而使飞船转移到半径为 $4 R$ 的轨道 3 上, 如图所示。若相距 $r$的两物体间引力势能为 $E_{\mathrm{p}}=-G \frac{M m}{r}$, 求: 飞船在轨道 2 上经过近地点 $A$ 和远地点 $B$ 的速率之比 $\left(v_{A}: v_{B}\right)$ 。 [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是数值。

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Astronomy

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multi-modal

Astronomy_1186

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. i. Determine the theoretical minimum angular diameter of an element resolvable by this optical system. Give your answer in arcseconds (").

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. i. Determine the theoretical minimum angular diameter of an element resolvable by this optical system. Give your answer in arcseconds ("). All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is an expression without equals signs, e.g. ANSWER=\frac{1}{2} g t^2

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Astronomy

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multi-modal

Astronomy_990

The year 1990 held many exciting events in astronomy. Which of the following did not celebrate its $30^{\text {th }}$ anniversary this year? [figure1] A: The launch of the Hubble Space Telescope, part of NASA's 'Great Observatories' programme B: The most distant photo ever taken of the Earth by the probe Voyager 1 from 40.5 au, nicknamed the 'Pale Blue Dot' by Carl Sagan C: The arrival at Venus of the Magellan probe, where it used radar to create the highest resolution maps we have of the surface of the planet D: ESA's Giotto probe passing within $600 \mathrm{~km}$ of the nucleus of Halley's Comet, taking our best photos of this famous object

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 year 1990 held many exciting events in astronomy. Which of the following did not celebrate its $30^{\text {th }}$ anniversary this year? [figure1] A: The launch of the Hubble Space Telescope, part of NASA's 'Great Observatories' programme B: The most distant photo ever taken of the Earth by the probe Voyager 1 from 40.5 au, nicknamed the 'Pale Blue Dot' by Carl Sagan C: The arrival at Venus of the Magellan probe, where it used radar to create the highest resolution maps we have of the surface of the planet D: ESA's Giotto probe passing within $600 \mathrm{~km}$ of the nucleus of Halley's Comet, taking our best photos of this famous object 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_705

《流浪地球 2 》中太空电梯非常吸引观众眼球。太空电梯通过超级缆绳连接地球赤道上的固定基地与配重空间站, 它们随地球以同步静止状态一起旋转，如图所示。图中配重空间站质量为 $m$, 比同步卫星更高, 距地面高达 $9 R$ 。若地球半径为 $R$, 自转周期为 $T$, 重力加速度为 $g$, 求: 若配重空间站没有缆绳连接, 在该处绕地球做匀速圆周运动的线速度大小为多少? [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: 《流浪地球 2 》中太空电梯非常吸引观众眼球。太空电梯通过超级缆绳连接地球赤道上的固定基地与配重空间站, 它们随地球以同步静止状态一起旋转，如图所示。图中配重空间站质量为 $m$, 比同步卫星更高, 距地面高达 $9 R$ 。若地球半径为 $R$, 自转周期为 $T$, 重力加速度为 $g$, 求: 若配重空间站没有缆绳连接, 在该处绕地球做匀速圆周运动的线速度大小为多少? [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_249

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 $g$; 单舱模式是指让固定在单舱上的实验平台随单舱在落塔中自由下落实现微重力环境。若舱体下落时, 受到的阻力恒为舱体总重力的 0.01 倍, 求单舱中的微重力加速度的大小 $g_{1} ;$ [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 $g$; 单舱模式是指让固定在单舱上的实验平台随单舱在落塔中自由下落实现微重力环境。若舱体下落时, 受到的阻力恒为舱体总重力的 0.01 倍, 求单舱中的微重力加速度的大小 $g_{1} ;$ [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

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Astronomy

ZH

multi-modal

Astronomy_269

如图所示为发射同步卫星的三个轨道, 轨道 I 为近地轨道, 轨道 II 为转移轨道, 轨道 III 为同步轨道, $P 、 Q$ 分别是转移轨道的近地点和远地点。假设卫星在各轨道运行时质量不变, 关于卫星在这个三个轨道上的运动, 下列说法正确的是 ( ) [图1] A: 卫星在各个轨道上的运行速度一定都小于 $7.9 \mathrm{~km} / \mathrm{s}$ B: 卫星在轨道 III 上 $Q$ 点的运行速度小于在轨道 II 上 $Q$ 点的运行速度 C: 卫星在轨道 II 上从 $P$ 点运动到 $Q$ 点的过程中，运行时间一定小于 $12 \mathrm{~h}$ D: 卫星在各个轨道上的机械能一样大

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: 如图所示为发射同步卫星的三个轨道, 轨道 I 为近地轨道, 轨道 II 为转移轨道, 轨道 III 为同步轨道, $P 、 Q$ 分别是转移轨道的近地点和远地点。假设卫星在各轨道运行时质量不变, 关于卫星在这个三个轨道上的运动, 下列说法正确的是 ( ) [图1] A: 卫星在各个轨道上的运行速度一定都小于 $7.9 \mathrm{~km} / \mathrm{s}$ B: 卫星在轨道 III 上 $Q$ 点的运行速度小于在轨道 II 上 $Q$ 点的运行速度 C: 卫星在轨道 II 上从 $P$ 点运动到 $Q$ 点的过程中，运行时间一定小于 $12 \mathrm{~h}$ D: 卫星在各个轨道上的机械能一样大 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

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multi-modal

Astronomy_930

From the upper parts of the Sun's corona comes a stream of charged particles called the solar wind, meaning the Sun is slowly losing some of its mass (although the effect is negligible compared to the mass loss in nuclear fusion). The particles travel at supersonic speeds until the pressure from interstellar space causes their speed to drop to subsonic speeds instead - this transition is called the termination shock, and has been explored by the two Voyager probes as they leave the solar system. [figure1] Figure 3: Left: A demonstration of a termination shock formed with water flowing from a tap into a sink. Right: The Voyager spacecraft crossing the termination shock of the Solar System. As the solar wind moves further from the Sun its speed increases (at an ever decreasing rate), until it asymptotes at a speed, $u_{\infty}$, equal to the escape velocity, $v_{\text {esc }}$, of the Sun. Given that $v_{\text {esc }}=\sqrt{2 G M / R}$ for an object with radius $R$ and mass $M$, calculate the escape velocity of the Sun. Give your answer in $\mathrm{km} \mathrm{s}^{-1}$.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. problem: From the upper parts of the Sun's corona comes a stream of charged particles called the solar wind, meaning the Sun is slowly losing some of its mass (although the effect is negligible compared to the mass loss in nuclear fusion). The particles travel at supersonic speeds until the pressure from interstellar space causes their speed to drop to subsonic speeds instead - this transition is called the termination shock, and has been explored by the two Voyager probes as they leave the solar system. [figure1] Figure 3: Left: A demonstration of a termination shock formed with water flowing from a tap into a sink. Right: The Voyager spacecraft crossing the termination shock of the Solar System. As the solar wind moves further from the Sun its speed increases (at an ever decreasing rate), until it asymptotes at a speed, $u_{\infty}$, equal to the escape velocity, $v_{\text {esc }}$, of the Sun. Given that $v_{\text {esc }}=\sqrt{2 G M / R}$ for an object with radius $R$ and mass $M$, calculate the escape velocity of the Sun. Give your answer in $\mathrm{km} \mathrm{s}^{-1}$. 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, 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_679

北京时间 2019 年 4 月 10 日晚 21 时 07 分人类首张黑洞照片面向全球同步发布（如图所示)。照片中间黑色的才是黑洞本体，直径大概 $1 \times 10^{11} \mathrm{~km}$, 周围是被它吸积成一圈的气体, 因湍急流动而摩擦发光。从地球上看, 这个距离我们 5000 万光年的 M87 黑洞是在顺时针旋转的。黑洞质量和半径 $R$ 的关系满足 $\frac{M}{R}=\frac{c^{2}}{2 G}$ (其中 $c$ 为光速, $G$ 为引力常量)，则该黑洞表面重力加速度的数量级为（） [图1] A: $9 \times 10^{2} \mathrm{~m} / \mathrm{s}^{2}$ B: $10^{5} \mathrm{~m} / \mathrm{s}^{2}$ C: $10 \mathrm{~m} / \mathrm{s}^{2}$ D: $10^{9} \mathrm{~m} / \mathrm{s}^{2}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 北京时间 2019 年 4 月 10 日晚 21 时 07 分人类首张黑洞照片面向全球同步发布（如图所示)。照片中间黑色的才是黑洞本体，直径大概 $1 \times 10^{11} \mathrm{~km}$, 周围是被它吸积成一圈的气体, 因湍急流动而摩擦发光。从地球上看, 这个距离我们 5000 万光年的 M87 黑洞是在顺时针旋转的。黑洞质量和半径 $R$ 的关系满足 $\frac{M}{R}=\frac{c^{2}}{2 G}$ (其中 $c$ 为光速, $G$ 为引力常量)，则该黑洞表面重力加速度的数量级为（） [图1] A: $9 \times 10^{2} \mathrm{~m} / \mathrm{s}^{2}$ B: $10^{5} \mathrm{~m} / \mathrm{s}^{2}$ C: $10 \mathrm{~m} / \mathrm{s}^{2}$ D: $10^{9} \mathrm{~m} / \mathrm{s}^{2}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

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multi-modal

Astronomy_647

下列物体的加速度最大的是（ ） A. 加速升空阶段的火箭 B. 月球上自由下落的物体 C. 击发后在枪筒中的子弹 D. 在地表随地球自转的物体 A: 加速升空阶段的火箭 B: 月球上自由下落的物体 C: 击发后在枪筒中的子弹 D: 在地表随地球自转的物体

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: 下列物体的加速度最大的是（ ） A. 加速升空阶段的火箭 B. 月球上自由下落的物体 C. 击发后在枪筒中的子弹 D. 在地表随地球自转的物体 A: 加速升空阶段的火箭 B: 月球上自由下落的物体 C: 击发后在枪筒中的子弹 D: 在地表随地球自转的物体 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_174

5 月 15 日, 我国首次火星探测任务天问一号探测器在火星乌托邦平原南部预选着陆区着陆, 在火星上首次留下中国印迹, 地球上每 26 个月才有一个火星探测器发射窗口, 此时出发最节省燃料。地球和火星绕太阳公转均可视为匀速圆周运动, 已知地球公转平均半径约 1.5 亿千米, 火星公转平均半径约 2.25 亿千米, 地球与火星处于何种位置关系时从地球发射天问一号能经粗圆轨道直接被火星捕获进入环火轨道运行（） A: [图1] B: [图2] C: [图3] D: [图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: 5 月 15 日, 我国首次火星探测任务天问一号探测器在火星乌托邦平原南部预选着陆区着陆, 在火星上首次留下中国印迹, 地球上每 26 个月才有一个火星探测器发射窗口, 此时出发最节省燃料。地球和火星绕太阳公转均可视为匀速圆周运动, 已知地球公转平均半径约 1.5 亿千米, 火星公转平均半径约 2.25 亿千米, 地球与火星处于何种位置关系时从地球发射天问一号能经粗圆轨道直接被火星捕获进入环火轨道运行（） A: [图1] B: [图2] C: [图3] D: [图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

multi-modal

Astronomy_984

Special Relativity (SR) tells us that two observers will disagree about the duration of a time interval measured by each one's clock if one is moving at speed $v$ relative to the other, a phenomenon called time dilation. General Relativity (GR) tells us that gravitational fields dilate time too. This has an impact on satellites, since they travel at high orbital speeds (slowing down their clocks relative to the surface) but due to their altitude they are in a weaker gravitational field (speeding up their clocks relative to the surface). Which effect is dominant varies with orbital radius. Global Positioning System (GPS) satellites must compensate for this effect, since the satellites rely on accurate measurements of the time between sending and receiving a radio signal. [figure1] Figure 4: A scale diagram of the positions of the orbits for the International Space Station (ISS), GPS satellites and geostationary satellites, along with their orbital periods In $\mathrm{SR}$, time dilation can be calculated with $$ t^{\prime}=\gamma t_{0} \quad \text { where } \quad \gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}} \quad \text { so } \quad \Delta t_{\mathrm{SR}}=t_{0}-t^{\prime}=(1-\gamma) t_{0} $$ where $t_{0}$ is the time measured by the moving clock, $t^{\prime}$ is the time measured by the observer, $c$ is the speed of light and $v$ is the speed of the object. A negative $\Delta t$ indicates that the clocks are passing time slower relative to the observer, whilst a positive indicates they are passing quicker. Given $a_{\text {GPS }}$ is the radius of the GPS satellite's orbit, calculate $\Delta t_{\text {overall }}$ when $t_{0}=1$ day. Give your answer in $\mu$ s and state the significance of the sign.

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: Special Relativity (SR) tells us that two observers will disagree about the duration of a time interval measured by each one's clock if one is moving at speed $v$ relative to the other, a phenomenon called time dilation. General Relativity (GR) tells us that gravitational fields dilate time too. This has an impact on satellites, since they travel at high orbital speeds (slowing down their clocks relative to the surface) but due to their altitude they are in a weaker gravitational field (speeding up their clocks relative to the surface). Which effect is dominant varies with orbital radius. Global Positioning System (GPS) satellites must compensate for this effect, since the satellites rely on accurate measurements of the time between sending and receiving a radio signal. [figure1] Figure 4: A scale diagram of the positions of the orbits for the International Space Station (ISS), GPS satellites and geostationary satellites, along with their orbital periods In $\mathrm{SR}$, time dilation can be calculated with $$ t^{\prime}=\gamma t_{0} \quad \text { where } \quad \gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}} \quad \text { so } \quad \Delta t_{\mathrm{SR}}=t_{0}-t^{\prime}=(1-\gamma) t_{0} $$ where $t_{0}$ is the time measured by the moving clock, $t^{\prime}$ is the time measured by the observer, $c$ is the speed of light and $v$ is the speed of the object. A negative $\Delta t$ indicates that the clocks are passing time slower relative to the observer, whilst a positive indicates they are passing quicker. Given $a_{\text {GPS }}$ is the radius of the GPS satellite's orbit, calculate $\Delta t_{\text {overall }}$ when $t_{0}=1$ day. Give your answer in $\mu$ s and state the significance of the sign. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of s, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

multi-modal

Astronomy_1172

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 $7.35 \times 10^{22} \mathrm{~kg}$.a. Ignoring the effects of air resistance, the weight of the rocket, and assuming 1-D motion only ii. Determine the constant acceleration produced by the third stage (S-IVB) of the Saturn V rocket and hence calculate the total mass carried into the parking orbit at the end of the first burn.

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

EX

Astronomy

EN

multi-modal

Astronomy_798

In 1995, researchers at the University of Geneva discovered an exoplanet in the main-sequence star 51 Pegasi. This was the first-ever discovery of an exoplanet orbiting a Sun-like star! When they observed the star, a periodic Doppler shifting of its stellar spectrum indicated that its radial velocity was varying sinusoidally; this wobbling could be explained if the star was being pulled in a circle by the gravity of an exoplanet. The radial velocity sinusoid of 51 Pegasi was measured to have a semi-amplitude of $56 \mathrm{~m} / \mathrm{s}$ with a period of 4.2 days, and the mass of the star is known to be $1.1 M_{\odot}$. Assuming that the researchers at Geneva viewed the planet's orbit edge-on and that the orbit was circular, what is the mass of the exoplanet in Jupiter masses? A: $0.81 M_{4}$ B: $0.75 M_{4}$ C: $0.69 M_{4}$ D: $0.47 M_{4}$ E: $0.33 M_{4}$

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 1995, researchers at the University of Geneva discovered an exoplanet in the main-sequence star 51 Pegasi. This was the first-ever discovery of an exoplanet orbiting a Sun-like star! When they observed the star, a periodic Doppler shifting of its stellar spectrum indicated that its radial velocity was varying sinusoidally; this wobbling could be explained if the star was being pulled in a circle by the gravity of an exoplanet. The radial velocity sinusoid of 51 Pegasi was measured to have a semi-amplitude of $56 \mathrm{~m} / \mathrm{s}$ with a period of 4.2 days, and the mass of the star is known to be $1.1 M_{\odot}$. Assuming that the researchers at Geneva viewed the planet's orbit edge-on and that the orbit was circular, what is the mass of the exoplanet in Jupiter masses? A: $0.81 M_{4}$ B: $0.75 M_{4}$ C: $0.69 M_{4}$ D: $0.47 M_{4}$ E: $0.33 M_{4}$ 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_819

After a day spent showing a visiting friend around Boston, Leo is walking back along the bridge (see diagram in previous problem) to return to Next House. The time is such that the Sun now aligns with perfectly upriver, so it is in the opposite direction compared to the morning. How high in the sky is the Sun relative to the morning? A: Above the horizon, at the same altitude as in the morning B: Above the horizon, but lower than in the morning C: On the horizon D: Below the horizon

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: After a day spent showing a visiting friend around Boston, Leo is walking back along the bridge (see diagram in previous problem) to return to Next House. The time is such that the Sun now aligns with perfectly upriver, so it is in the opposite direction compared to the morning. How high in the sky is the Sun relative to the morning? A: Above the horizon, at the same altitude as in the morning B: Above the horizon, but lower than in the morning C: On the horizon D: Below the horizon You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

SC

Astronomy

EN

text-only

Astronomy_719

如图所示, 微信启动新界面, 其画面视角从非洲大陆上空（左）变成中国上空 （右），新照片由我国新一代静止轨道卫星“风云四号”拍摄，见证着科学家 15 年的辛苦和努力。下列说法正确的是（ ） [图1] A: “风云四号”可能经过北京正上空 B: “风云四号”不可能经过北京正上空 C: 与“风云四号”同轨道的卫星运动的动能都相等 D: “风云四号”的运行速度大于 $7.9 \mathrm{~km} / \mathrm{s}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 如图所示, 微信启动新界面, 其画面视角从非洲大陆上空（左）变成中国上空 （右），新照片由我国新一代静止轨道卫星“风云四号”拍摄，见证着科学家 15 年的辛苦和努力。下列说法正确的是（ ） [图1] A: “风云四号”可能经过北京正上空 B: “风云四号”不可能经过北京正上空 C: 与“风云四号”同轨道的卫星运动的动能都相等 D: “风云四号”的运行速度大于 $7.9 \mathrm{~km} / \mathrm{s}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

multi-modal

Astronomy_750

What is the name of the JWST component highlighted below? [figure1] A: Antenna B: Sunshield C: Optics subsystem D: Stabilization flap

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 the JWST component highlighted below? [figure1] A: Antenna B: Sunshield C: Optics subsystem D: Stabilization flap 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_898

As of September 2021, which of the following billionaires has not been into space (defined as at least $80 \mathrm{~km}$ above the surface of the Earth)? [figure1] A: Jeff Bezos with the company Blue Origin B: Richard Branson with the company Virgin Galactic C: Elon Musk with the company SpaceX D: Dennis Tito with the company Space Adventures

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: As of September 2021, which of the following billionaires has not been into space (defined as at least $80 \mathrm{~km}$ above the surface of the Earth)? [figure1] A: Jeff Bezos with the company Blue Origin B: Richard Branson with the company Virgin Galactic C: Elon Musk with the company SpaceX D: Dennis Tito with the company Space Adventures 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_488

2021 年 10 月 16 日, 神舟十三号载人飞船采用自主快速交会对接模式成功对接于天和核心舱径向端口，对接过程简化如图所示。神舟十三号先到达天和核心舱轨道正下方 $d_{l}=200$ 米的第一停泊点并保持相对静止, 完成各种测控后, 开始沿地心与天和核心舱连线 (径向) 向天和核心舱靠近, 到距离天和核心舱 $d_{2}=19$ 米的第二停泊点短暂驻留,完成各种测控后, 继续径向靠近, 以很小的相对速度完成精准的端口对接。对接技术非常复杂, 故做如下简化。假设地球是半径为 $R_{0}$ 的标准球体，地表重力加速度为 $g$, 忽略自转; 核心舱轨道是半径为 $R$ 的正圆; 神舟十三号质量为 $m_{l}$, 对接前组合体的总质量为 $m_{2}$ ；忽略对接前后神舟十三号质量的变化。 虽然对接时两者相对速度很小，但如果不及时控制也会造成组合体偏离正确轨道,假设不考虑转动, 设对接靠近速度为 $v$, 求控制组合体轨道复位的火箭要对组合体做的功 $W$ 。 [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: 2021 年 10 月 16 日, 神舟十三号载人飞船采用自主快速交会对接模式成功对接于天和核心舱径向端口，对接过程简化如图所示。神舟十三号先到达天和核心舱轨道正下方 $d_{l}=200$ 米的第一停泊点并保持相对静止, 完成各种测控后, 开始沿地心与天和核心舱连线 (径向) 向天和核心舱靠近, 到距离天和核心舱 $d_{2}=19$ 米的第二停泊点短暂驻留,完成各种测控后, 继续径向靠近, 以很小的相对速度完成精准的端口对接。对接技术非常复杂, 故做如下简化。假设地球是半径为 $R_{0}$ 的标准球体，地表重力加速度为 $g$, 忽略自转; 核心舱轨道是半径为 $R$ 的正圆; 神舟十三号质量为 $m_{l}$, 对接前组合体的总质量为 $m_{2}$ ；忽略对接前后神舟十三号质量的变化。 虽然对接时两者相对速度很小，但如果不及时控制也会造成组合体偏离正确轨道,假设不考虑转动, 设对接靠近速度为 $v$, 求控制组合体轨道复位的火箭要对组合体做的功 $W$ 。 [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_61

宇宙中有许多双星系统由两颗恒星组成, 两恒星在相互引力的作用下, 分别围绕其连线上的某一点做周期相同的匀速圆周运动. 研究发现, 双星系统演化过程中，两星的总质量、距离和周期均可能发生变化. 若某双星系统中两星做圆周运动的周期为 $\mathrm{T}, \mathrm{M}_{1}$星线速度大小为 $\mathrm{v}_{1}, \mathrm{M}_{2}$ 星线速度大小为 $\mathrm{v}_{2}$, 经过一段时间演化后, 两星总质量变为原来的 $\frac{1}{k}(\mathrm{k}>1)$ 倍, 两星之间的距离变为原来的 $\mathrm{n}(\mathrm{n}>1)$ 倍, 则此时双星系统圆周运动的周期 $\mathrm{T}^{\prime}$ 和线速度之和 $\mathrm{v}_{1}{ }^{\prime}+\mathrm{v}_{2}{ }^{\prime}$ 是 [图1] A: $T^{\prime}=\sqrt{n^{3} k} T$ B: $T^{\prime}=\sqrt{\frac{n^{3}}{k}} T$ C: $v_{1}^{\prime}+v_{2}^{\prime}=\frac{1}{\sqrt{n k}}\left(v_{1}+v_{2}\right)$ D: $v_{1}^{\prime}+v_{2}^{\prime}=\sqrt{n k}\left(v_{1}+v_{2}\right)$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 宇宙中有许多双星系统由两颗恒星组成, 两恒星在相互引力的作用下, 分别围绕其连线上的某一点做周期相同的匀速圆周运动. 研究发现, 双星系统演化过程中，两星的总质量、距离和周期均可能发生变化. 若某双星系统中两星做圆周运动的周期为 $\mathrm{T}, \mathrm{M}_{1}$星线速度大小为 $\mathrm{v}_{1}, \mathrm{M}_{2}$ 星线速度大小为 $\mathrm{v}_{2}$, 经过一段时间演化后, 两星总质量变为原来的 $\frac{1}{k}(\mathrm{k}>1)$ 倍, 两星之间的距离变为原来的 $\mathrm{n}(\mathrm{n}>1)$ 倍, 则此时双星系统圆周运动的周期 $\mathrm{T}^{\prime}$ 和线速度之和 $\mathrm{v}_{1}{ }^{\prime}+\mathrm{v}_{2}{ }^{\prime}$ 是 [图1] A: $T^{\prime}=\sqrt{n^{3} k} T$ B: $T^{\prime}=\sqrt{\frac{n^{3}}{k}} T$ C: $v_{1}^{\prime}+v_{2}^{\prime}=\frac{1}{\sqrt{n k}}\left(v_{1}+v_{2}\right)$ D: $v_{1}^{\prime}+v_{2}^{\prime}=\sqrt{n k}\left(v_{1}+v_{2}\right)$ 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_42

中国计划已经实现返回式月球软着陆器对月球进行科学探测, 如图所示, 发射一颗运动半径为 $r$ 的绕月卫星, 登月着陆器从绕月卫星出发 (不影响绕月卫星运动), 沿粗圆轨道降落到月球的表面上, 与月球表面经多次碰撞和弹跳才停下来。假设着陆器第一次弹起的最大高度为 $h$, 水平速度为 $v_{1}$, 第二次着陆时速度为 $v_{2}$, 已知月球半径为 $R$,着陆器质量为 $m$, 不计一切阻力和月球的自转。求: 设想软着陆器完成了对月球的科学考察任务后, 再返回绕月卫星, 返回与卫星对接时, 二者具有相同的速度, 着陆器在返回过程中需克服月球引力做功 $W=m g_{\text {月 }}\left(1-\frac{R}{r}\right) R$, 则着陆器的电池应提供给着陆器多少能量, 才能使着陆器安全返回到绕月卫星。 [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: 中国计划已经实现返回式月球软着陆器对月球进行科学探测, 如图所示, 发射一颗运动半径为 $r$ 的绕月卫星, 登月着陆器从绕月卫星出发 (不影响绕月卫星运动), 沿粗圆轨道降落到月球的表面上, 与月球表面经多次碰撞和弹跳才停下来。假设着陆器第一次弹起的最大高度为 $h$, 水平速度为 $v_{1}$, 第二次着陆时速度为 $v_{2}$, 已知月球半径为 $R$,着陆器质量为 $m$, 不计一切阻力和月球的自转。求: 设想软着陆器完成了对月球的科学考察任务后, 再返回绕月卫星, 返回与卫星对接时, 二者具有相同的速度, 着陆器在返回过程中需克服月球引力做功 $W=m g_{\text {月 }}\left(1-\frac{R}{r}\right) R$, 则着陆器的电池应提供给着陆器多少能量, 才能使着陆器安全返回到绕月卫星。 [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_251

人造卫星在绕地球运行时, 会遇到稀薄大气的阻力。如果不进行必要的轨道维持,稀薄大气对卫星的这种微小阻力会导致卫星轨道半径逐渐减小, 以至最终落回地球。这个过程是非常漫长的, 因此卫星每一圈的运动仍可以认为是匀速圆周运动。规定两质点相距无穷远时的引力势能为零, 理论上可以得出质量分别 $m_{1} 、 m_{2}$ 的两个物体相距 $r$ 时,系统的引力势能为 $E_{p}=\frac{G m_{1} m_{2}}{r}$ 。已知人造卫星的质量为 $m$, 某时刻绕地球做匀速圆周运动的轨道半径为 $r$, 地球半径为 $R$, 地球表面附近的重力加速度为 $g$ 。 由于大气阻力的影响, 卫星的轨道半径逐渐减小。求在这个过程中, 万有引力做的功 $\mathrm{W}_{\mathrm{G}}$ 与克服大气阻力做的功 $\mathrm{W}_{\mathrm{f}}$ 的比。

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个数值。 问题: 人造卫星在绕地球运行时, 会遇到稀薄大气的阻力。如果不进行必要的轨道维持,稀薄大气对卫星的这种微小阻力会导致卫星轨道半径逐渐减小, 以至最终落回地球。这个过程是非常漫长的, 因此卫星每一圈的运动仍可以认为是匀速圆周运动。规定两质点相距无穷远时的引力势能为零, 理论上可以得出质量分别 $m_{1} 、 m_{2}$ 的两个物体相距 $r$ 时,系统的引力势能为 $E_{p}=\frac{G m_{1} m_{2}}{r}$ 。已知人造卫星的质量为 $m$, 某时刻绕地球做匀速圆周运动的轨道半径为 $r$, 地球半径为 $R$, 地球表面附近的重力加速度为 $g$ 。 由于大气阻力的影响, 卫星的轨道半径逐渐减小。求在这个过程中, 万有引力做的功 $\mathrm{W}_{\mathrm{G}}$ 与克服大气阻力做的功 $\mathrm{W}_{\mathrm{f}}$ 的比。 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是数值。

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Astronomy

ZH

text-only

Astronomy_615

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。 求空间站绕地球匀速圆周运动的动能块 $E_{\mathrm{k} 1}$;

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 2022 年 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。 求空间站绕地球匀速圆周运动的动能块 $E_{\mathrm{k} 1}$; 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

text-only

Astronomy_731

某卫星绕地球做匀速圆周运动, 地球相对卫星的张角为 $\theta$, 当卫星与地心连线扫过 $\theta$ (弧度) 的角度时, 圆周运动通过的弧长为 $s$, 已知地球表面的重力加速度为 $g$, 则下列判断正确的是 ( )。 A: 地球半径为 $\frac{s}{\theta} \cos \frac{\theta}{2}$ B: 卫星的向心加速度为 $g \sin \frac{\theta}{2}$ C: 卫星的线速度为 $\sqrt{\frac{2 g s}{\theta}} \cdot \sin \frac{\theta}{2}$ D: 卫星的角速度为 $\sqrt{\frac{g \theta}{s}} \cdot \sin \frac{\theta}{2}$

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: 某卫星绕地球做匀速圆周运动, 地球相对卫星的张角为 $\theta$, 当卫星与地心连线扫过 $\theta$ (弧度) 的角度时, 圆周运动通过的弧长为 $s$, 已知地球表面的重力加速度为 $g$, 则下列判断正确的是 ( )。 A: 地球半径为 $\frac{s}{\theta} \cos \frac{\theta}{2}$ B: 卫星的向心加速度为 $g \sin \frac{\theta}{2}$ C: 卫星的线速度为 $\sqrt{\frac{2 g s}{\theta}} \cdot \sin \frac{\theta}{2}$ D: 卫星的角速度为 $\sqrt{\frac{g \theta}{s}} \cdot \sin \frac{\theta}{2}$ 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_481

地球的公转轨道接近圆, 哈雷彗星的公转轨道则是一个非常扁的粗圆, 如图所示。天文学家哈雷成功预言了哈雷彗星的回归, 哈雷彗星最近出现的时间是 1986 年, 预测下次飞近地球将在 2061 年左右。若哈雷彗星在近日点与太阳中心的距离为 $r_{1}$, 远日点与太阳中心的距离为 $r_{2}$ 。下列说法正确的是 ( ) [图1] A: 哈雷彗星轨道的半长轴是地球公转半径的 $\sqrt{75^{3}}$ 倍 B: 哈雷彗星在近日点的速度一定大于地球的公转速度 C: 哈雷彗星在近日点和远日点的速度之比为 $\sqrt{r_{2}}: \sqrt{r_{1}}$ D: 相同时间内, 哈雷彗星与太阳连线扫过的面积和地球与太阳连线扫过的面积相等

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 地球的公转轨道接近圆, 哈雷彗星的公转轨道则是一个非常扁的粗圆, 如图所示。天文学家哈雷成功预言了哈雷彗星的回归, 哈雷彗星最近出现的时间是 1986 年, 预测下次飞近地球将在 2061 年左右。若哈雷彗星在近日点与太阳中心的距离为 $r_{1}$, 远日点与太阳中心的距离为 $r_{2}$ 。下列说法正确的是 ( ) [图1] A: 哈雷彗星轨道的半长轴是地球公转半径的 $\sqrt{75^{3}}$ 倍 B: 哈雷彗星在近日点的速度一定大于地球的公转速度 C: 哈雷彗星在近日点和远日点的速度之比为 $\sqrt{r_{2}}: \sqrt{r_{1}}$ D: 相同时间内, 哈雷彗星与太阳连线扫过的面积和地球与太阳连线扫过的面积相等 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

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multi-modal

Astronomy_416

2019 年 12 月 17 日, 中国科学院云南天文台研究人员在对密近双星半人马座 V752 进行观测和分析研究时, 发现了一种双星轨道变化的新模式。双星在运行时周期突然增大, 研究人员分析有可能是受到了来自其伴星双星的动力学扰动, 从而引起了双星间的物质相互交流，周期开始持续增加。若小质量的子星的物质被吸引而转移至大质量的子星上 (不考虑质量的损失), 导致周期增大为原来的 $k$ 倍, 则下列说法中正确的是 ( ) A: 两子星间距增大为原来的 $k^{\frac{3}{2}}$ 倍 B: 两子星间的万有引力增大 C: 质量小的子星轨道半径增大 D: 质量大的子星线速度增大

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 2019 年 12 月 17 日, 中国科学院云南天文台研究人员在对密近双星半人马座 V752 进行观测和分析研究时, 发现了一种双星轨道变化的新模式。双星在运行时周期突然增大, 研究人员分析有可能是受到了来自其伴星双星的动力学扰动, 从而引起了双星间的物质相互交流，周期开始持续增加。若小质量的子星的物质被吸引而转移至大质量的子星上 (不考虑质量的损失), 导致周期增大为原来的 $k$ 倍, 则下列说法中正确的是 ( ) A: 两子星间距增大为原来的 $k^{\frac{3}{2}}$ 倍 B: 两子星间的万有引力增大 C: 质量小的子星轨道半径增大 D: 质量大的子星线速度增大 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

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text-only

Astronomy_466

科幻影片《流浪地球》中为了让地球逃离太阳系, 人们在地球上建造特大功率发动机, 使地球完成一系列变轨操作, 其逃离过程可设想成如图所示, 地球在椭圆轨道 I 上运行到远日点 $P$ 变轨进入圆形轨道 II, 在圆形轨道 II 上运行一段时间后在 $P$ 点时再次加速变轨, 从而最终摆脱太阳束缚。对于该过程, 下列说法正确的是( ) [图1] A: 地球在 $P$ 点通过向前喷气减速实现由轨道 I 进入轨道 II B: 若地球在 I、II 轨道上运行的周期分别为 $T_{1} 、 T_{2}$, 则 $T_{1}<T_{2}$, C: 地球在轨道 I 正常运行时(不含变轨时刻)经过 $P$ 点的加速度比地球在轨道 II 正常运行(不含变轨时刻)时经过 $P$ 点的加速度大 D: 地球在轨道 I 上过 $O$ 点的速率比地球在轨道 II 上过 $P$ 点的速率小

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: 科幻影片《流浪地球》中为了让地球逃离太阳系, 人们在地球上建造特大功率发动机, 使地球完成一系列变轨操作, 其逃离过程可设想成如图所示, 地球在椭圆轨道 I 上运行到远日点 $P$ 变轨进入圆形轨道 II, 在圆形轨道 II 上运行一段时间后在 $P$ 点时再次加速变轨, 从而最终摆脱太阳束缚。对于该过程, 下列说法正确的是( ) [图1] A: 地球在 $P$ 点通过向前喷气减速实现由轨道 I 进入轨道 II B: 若地球在 I、II 轨道上运行的周期分别为 $T_{1} 、 T_{2}$, 则 $T_{1}<T_{2}$, C: 地球在轨道 I 正常运行时(不含变轨时刻)经过 $P$ 点的加速度比地球在轨道 II 正常运行(不含变轨时刻)时经过 $P$ 点的加速度大 D: 地球在轨道 I 上过 $O$ 点的速率比地球在轨道 II 上过 $P$ 点的速率小 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

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multi-modal