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GAIR 66

Astronomy_257

迷你系绳卫星在地球赤道正上方的电离层中, 沿圆形轨道绕地飞行。系绳卫星由两子卫星组成, 它们之间的导体绳沿地球半径方向, 如图所示。在电池和感应电动势的共同作用下, 导体绳中形成指向地心的电流，等效总电阻为 $r$ 。导体绳所受的安培力克服大小为 $f$ 的环境阻力, 可使卫星保持在原轨道上。已知卫星离地平均高度为 $H$, 导体绳长为 $L(L \ll H)$, 地球半径为 $R$, 质量为 $M$, 万有引力常量为 $G$, 轨道处磁感应强度大小为 $B$, 方向垂直于导体绳。忽略地球自转的影响。求: 电池电动势 $E_{2}$ 。 [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: 迷你系绳卫星在地球赤道正上方的电离层中, 沿圆形轨道绕地飞行。系绳卫星由两子卫星组成, 它们之间的导体绳沿地球半径方向, 如图所示。在电池和感应电动势的共同作用下, 导体绳中形成指向地心的电流，等效总电阻为 $r$ 。导体绳所受的安培力克服大小为 $f$ 的环境阻力, 可使卫星保持在原轨道上。已知卫星离地平均高度为 $H$, 导体绳长为 $L(L \ll H)$, 地球半径为 $R$, 质量为 $M$, 万有引力常量为 $G$, 轨道处磁感应强度大小为 $B$, 方向垂直于导体绳。忽略地球自转的影响。求: 电池电动势 $E_{2}$ 。 [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_124

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 $T$, 经过一段时间演化后, 两星总质量变为原来的 $k$ 倍, 两星之间的距离变为原来的 $n$ 倍, 则 $(\quad)$ A: 经过一段时间演化后，其周期变为 $\sqrt{\frac{n^{3}}{k}} T$ B: 经过一段时间演化后，其角速度变为 $\frac{2 \pi}{T} \sqrt{\frac{k}{n^{2}}}$ C: 双星的轨道半径与各自质量成正比 D: 双星系统总质量、距离和周期未变化前, 双星中质量大的星体动能小, 质量小的星体动能大

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 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 $T$, 经过一段时间演化后, 两星总质量变为原来的 $k$ 倍, 两星之间的距离变为原来的 $n$ 倍, 则 $(\quad)$ A: 经过一段时间演化后，其周期变为 $\sqrt{\frac{n^{3}}{k}} T$ B: 经过一段时间演化后，其角速度变为 $\frac{2 \pi}{T} \sqrt{\frac{k}{n^{2}}}$ C: 双星的轨道半径与各自质量成正比 D: 双星系统总质量、距离和周期未变化前, 双星中质量大的星体动能小, 质量小的星体动能大 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

text-only

Astronomy_110

据报道美国人马斯克的“星链”系统在俄乌战争中发挥了重要作用。为此我国也开始规划自己的“星链”式卫星星座。据悉中国“星链”式卫星星座计划采用超低轨道。已知球冠面积公式 $S=2 \pi R h$ ( $\mathrm{R}$ 为球体半径、 $h$ 为球冠高度), 一颗超低轨道卫星绕地球做匀速圆周运动, 某时刻与地面通讯时能覆盖地球球冠的最大面积为 $S$, 已知地球半径为 $R_{1}$, 地球表面重力加速度为 $\mathrm{g}$ ，则下列选项中正确的是（） A: 该卫星绕地球运行的速度大于 $11.9 \mathrm{~km} / \mathrm{s}$ B: 该卫星距离地球表面的高度为 $\frac{S R_{1}}{2 \pi R_{1}{ }^{2}-S}$ C: 该卫星距离地球表面的高度为 $\frac{2 \pi R_{1}^{3}}{2 \pi R_{1}{ }^{2}-S}$ D: 若卫星做圆周运动的轨道半径为 $r$, 则运行周期为 $2 \pi \sqrt{\frac{r^{3}}{\mathrm{~g} R_{1}}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 据报道美国人马斯克的“星链”系统在俄乌战争中发挥了重要作用。为此我国也开始规划自己的“星链”式卫星星座。据悉中国“星链”式卫星星座计划采用超低轨道。已知球冠面积公式 $S=2 \pi R h$ ( $\mathrm{R}$ 为球体半径、 $h$ 为球冠高度), 一颗超低轨道卫星绕地球做匀速圆周运动, 某时刻与地面通讯时能覆盖地球球冠的最大面积为 $S$, 已知地球半径为 $R_{1}$, 地球表面重力加速度为 $\mathrm{g}$ ，则下列选项中正确的是（） A: 该卫星绕地球运行的速度大于 $11.9 \mathrm{~km} / \mathrm{s}$ B: 该卫星距离地球表面的高度为 $\frac{S R_{1}}{2 \pi R_{1}{ }^{2}-S}$ C: 该卫星距离地球表面的高度为 $\frac{2 \pi R_{1}^{3}}{2 \pi R_{1}{ }^{2}-S}$ D: 若卫星做圆周运动的轨道半径为 $r$, 则运行周期为 $2 \pi \sqrt{\frac{r^{3}}{\mathrm{~g} R_{1}}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

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text-only

Astronomy_1206

e. Calculate the projected physical separation, $r_{p}$, between the galaxy and the Voorwerp.f. Derive an expression for the difference in the light travel time between photons travelling directly to Earth from the galaxy and photons reflected off the Voorwerp first. Give your formula as a function of $r_{p}$ and $\theta$, where $\theta$ is the angle between the lines of sight to the Earth and to the centre of the Voorwerp as measured by an observer at the centre of IC 2497. (For example $\theta=90^{\circ}$ would correspond to the galaxy and Voorwerp both being the exact same distance from the Earth, and so the projected distance $r_{p}$ is therefore also the true distance between them.)

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: e. Calculate the projected physical separation, $r_{p}$, between the galaxy and the Voorwerp. problem: f. Derive an expression for the difference in the light travel time between photons travelling directly to Earth from the galaxy and photons reflected off the Voorwerp first. Give your formula as a function of $r_{p}$ and $\theta$, where $\theta$ is the angle between the lines of sight to the Earth and to the centre of the Voorwerp as measured by an observer at the centre of IC 2497. (For example $\theta=90^{\circ}$ would correspond to the galaxy and Voorwerp both being the exact same distance from the Earth, and so the projected distance $r_{p}$ is therefore also the true distance between them.) 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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text-only

Astronomy_786

Where in space is the JWST located? A: Geostationary orbit B: In the Moon's shadow C: Lagrange point D: Between Earth and Moon

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: Where in space is the JWST located? A: Geostationary orbit B: In the Moon's shadow C: Lagrange point D: Between Earth and Moon 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_643

地球同步卫星的加速度为 $a_{1}$, 运行速度为 $v_{1}$, 地面附近卫星的加速度为 $a_{2}$, 运行速度为 $v_{2}$, 地球赤道上物体随地球自转的向心加速度为 $a_{3}$, 运行速度为 $v_{3}$, 则 ( ) A: $v_{2}>v_{1}>v_{3}$ B: $v_{3}>v_{1}>v_{2}$ C: $a_{2}>a_{3}>a_{1}$ D: $a_{1}>a_{2}>a_{3}$

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: 地球同步卫星的加速度为 $a_{1}$, 运行速度为 $v_{1}$, 地面附近卫星的加速度为 $a_{2}$, 运行速度为 $v_{2}$, 地球赤道上物体随地球自转的向心加速度为 $a_{3}$, 运行速度为 $v_{3}$, 则 ( ) A: $v_{2}>v_{1}>v_{3}$ B: $v_{3}>v_{1}>v_{2}$ C: $a_{2}>a_{3}>a_{1}$ D: $a_{1}>a_{2}>a_{3}$ 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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Astronomy

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text-only

Astronomy_463

宇航员在某星球表面 (无空气) 将小球从空中坚直向下抛出, 测得小球速率的二次方与其离开抛出点的距离的关系如图所示 (图中的 $b 、 c 、 d$ 均为已知量)。该星球的半径为 $R$, 引力常量为 $G$, 将该星球视为球体, 忽略该星球的自转。下列说法正确的是 [图1] A: 该星球表面的重力加速度大小为 $\frac{c-b}{2 d}$ B: 该星球的质量为 $\frac{c R^{2}}{2 G d}$ C: 该星球的平均密度为 $\frac{3(c-b)}{8 \pi G d R}$ D: 该星球的第一宇宙速度为 $\sqrt{\frac{(c-b) R}{d}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 宇航员在某星球表面 (无空气) 将小球从空中坚直向下抛出, 测得小球速率的二次方与其离开抛出点的距离的关系如图所示 (图中的 $b 、 c 、 d$ 均为已知量)。该星球的半径为 $R$, 引力常量为 $G$, 将该星球视为球体, 忽略该星球的自转。下列说法正确的是 [图1] A: 该星球表面的重力加速度大小为 $\frac{c-b}{2 d}$ B: 该星球的质量为 $\frac{c R^{2}}{2 G d}$ C: 该星球的平均密度为 $\frac{3(c-b)}{8 \pi G d R}$ D: 该星球的第一宇宙速度为 $\sqrt{\frac{(c-b) R}{d}}$ 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_539

假设地球可视为质量均匀分布的球体。已知地球表面的重力加速度在两极的大小为 $g_{0}$, 在赤道的大小为 $g$; 地球半径为 $R$, 引力常数为 $G$, 则 ( ) A: 地球同步卫星距地表的高度为 $\left(\sqrt{\frac{g_{0}}{g_{0}-g}}-1\right) R$ B: 地球的质量为 $\frac{g_{0} R^{2}}{G}$ C: 地球的第一宇宙速度为 $\sqrt{g R}$ D: 地球密度为 $\frac{3 g}{4 \pi R G}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 假设地球可视为质量均匀分布的球体。已知地球表面的重力加速度在两极的大小为 $g_{0}$, 在赤道的大小为 $g$; 地球半径为 $R$, 引力常数为 $G$, 则 ( ) A: 地球同步卫星距地表的高度为 $\left(\sqrt{\frac{g_{0}}{g_{0}-g}}-1\right) R$ B: 地球的质量为 $\frac{g_{0} R^{2}}{G}$ C: 地球的第一宇宙速度为 $\sqrt{g R}$ D: 地球密度为 $\frac{3 g}{4 \pi R G}$ 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应为以下选项之一：[A, B, C, D]

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Astronomy

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text-only

Astronomy_1127

A "supermoon" is a new or full moon that occurs with the Moon at or near its closest approach to Earth in a given orbit (perigee). The media commonly associates supermoons with extreme brightness and size, sometimes implying that the Moon itself will become larger and have an impact on human behaviour, but just how different is a supermoon compared to the 'normal' Moon we see each month? Lunar Data: Synodic Period Anomalistic Period Semi-major axis Orbit eccentricity $$ \begin{aligned} & =29.530589 \text { days (time between same phases e.g. full moon to full moon) } \\ & =27.554550 \text { days (time between perigees i.e. perigee to perigee) } \\ & =3.844 \times 10^{5} \mathrm{~km} \\ & =0.0549 \\ & =1738.1 \mathrm{~km} \end{aligned} $$ $$ \begin{array}{ll} \text { Radius of the Moon } & =1738.1 \mathrm{~km} \\ \text { Mass of the Moon } & =7.342 \times 10^{22} \mathrm{~kg} \end{array} $$ In this question, we will only consider a full moon that is at perigee to be a supermoon.c). Determine the difference in the angular diameter of a supermoon and a full moon observed at apogee. Thus, determine the percentage difference in the brightness of a supermoon and a full moon observed at apogee. (Ignore the effects of the Moon's orbital tilt with respect to the Earth).

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: A "supermoon" is a new or full moon that occurs with the Moon at or near its closest approach to Earth in a given orbit (perigee). The media commonly associates supermoons with extreme brightness and size, sometimes implying that the Moon itself will become larger and have an impact on human behaviour, but just how different is a supermoon compared to the 'normal' Moon we see each month? Lunar Data: Synodic Period Anomalistic Period Semi-major axis Orbit eccentricity $$ \begin{aligned} & =29.530589 \text { days (time between same phases e.g. full moon to full moon) } \\ & =27.554550 \text { days (time between perigees i.e. perigee to perigee) } \\ & =3.844 \times 10^{5} \mathrm{~km} \\ & =0.0549 \\ & =1738.1 \mathrm{~km} \end{aligned} $$ $$ \begin{array}{ll} \text { Radius of the Moon } & =1738.1 \mathrm{~km} \\ \text { Mass of the Moon } & =7.342 \times 10^{22} \mathrm{~kg} \end{array} $$ In this question, we will only consider a full moon that is at perigee to be a supermoon. problem: c). Determine the difference in the angular diameter of a supermoon and a full moon observed at apogee. Thus, determine the percentage difference in the brightness of a supermoon and a full moon observed at apogee. (Ignore the effects of the Moon's orbital tilt with respect to the Earth). All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of % percentage, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

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Astronomy

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Astronomy_724

某国际研究小组观测到了一组双星系统, 它们绕二者连线上的某点做匀速圆周运动,双星系统中质量较小的星体能“吸食”质量较大的星体的表面物质，达到质量转移的目的. 根据大爆宇宙学可知, 双星间的距离在缓慢增大, 假设星体的轨道近似为圆, 则在该过程中（ ) A: 双星做圆周运动的角速度不断减小 B: 双星做圆周运动的角速度不断增大 C: 质量较大的星体做圆周运动的轨道半径减小 D: 质量较大的星体做圆周运动的轨道半径增大

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: 某国际研究小组观测到了一组双星系统, 它们绕二者连线上的某点做匀速圆周运动,双星系统中质量较小的星体能“吸食”质量较大的星体的表面物质，达到质量转移的目的. 根据大爆宇宙学可知, 双星间的距离在缓慢增大, 假设星体的轨道近似为圆, 则在该过程中（ ) A: 双星做圆周运动的角速度不断减小 B: 双星做圆周运动的角速度不断增大 C: 质量较大的星体做圆周运动的轨道半径减小 D: 质量较大的星体做圆周运动的轨道半径增大 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]

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Astronomy

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text-only

Astronomy_817

Billions of years from now, as the Moon moves farther away from the Earth, the Earth's axial tilt may become unstable. Imagine the Earth's tilt is such that the angle between the celestial equator and the ecliptic is $60^{\circ}$, rather than the current $23.44^{\circ}$ - so the Arctic Circle is now as far south as $30^{\circ}$ North. For an observer at $40^{\circ}$ North, how many days out of the year would the Sun never set (also known as the "polar day")? (Ignore atmospheric refraction, and assume the Earth's orbit is circular and nothing else has changed from today.) A: 28 B: 56 C: 61 D: 67 E: 113

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: Billions of years from now, as the Moon moves farther away from the Earth, the Earth's axial tilt may become unstable. Imagine the Earth's tilt is such that the angle between the celestial equator and the ecliptic is $60^{\circ}$, rather than the current $23.44^{\circ}$ - so the Arctic Circle is now as far south as $30^{\circ}$ North. For an observer at $40^{\circ}$ North, how many days out of the year would the Sun never set (also known as the "polar day")? (Ignore atmospheric refraction, and assume the Earth's orbit is circular and nothing else has changed from today.) A: 28 B: 56 C: 61 D: 67 E: 113 You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D, E].

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Astronomy

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Astronomy_753

Compared to the Sun's surface temperature, sunspots are ... A: cooler B: hotter C: same temperature D: sometimes hotter and sometimes cooler

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: Compared to the Sun's surface temperature, sunspots are ... A: cooler B: hotter C: same temperature D: sometimes hotter and sometimes cooler 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_1207

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.c. Computer models suggest the first galaxies formed around $z \sim 10-20$. One of the best ways to look for high-redshift galaxies is to try and detect the emission from the Lyman alpha (Lya) emission line at $\lambda_{\text {emit }}=121.6 \mathrm{~nm}$ as it is a relatively bright line. Some of the brightest galaxies in that initial era of galaxy formation would have an absolute magnitude of $\mathcal{M} \sim 20$. In this question, you are given that $\Omega_{0, \mathrm{~m}}=0.3, \Omega_{0, \Lambda}=0.7, \Omega_{0, \mathrm{r}}=0$ and $\mathrm{H}_{0}=70 \mathrm{~km} \mathrm{~s}^{-1} \mathrm{Mpc}^{-1}$. iii. Calculate the luminosity distance to the galaxy and hence its apparent magnitude. Assume all emitted flux is picked up by the telescope.

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: c. Computer models suggest the first galaxies formed around $z \sim 10-20$. One of the best ways to look for high-redshift galaxies is to try and detect the emission from the Lyman alpha (Lya) emission line at $\lambda_{\text {emit }}=121.6 \mathrm{~nm}$ as it is a relatively bright line. Some of the brightest galaxies in that initial era of galaxy formation would have an absolute magnitude of $\mathcal{M} \sim 20$. In this question, you are given that $\Omega_{0, \mathrm{~m}}=0.3, \Omega_{0, \Lambda}=0.7, \Omega_{0, \mathrm{r}}=0$ and $\mathrm{H}_{0}=70 \mathrm{~km} \mathrm{~s}^{-1} \mathrm{Mpc}^{-1}$. iii. Calculate the luminosity distance to the galaxy and hence its apparent magnitude. Assume all emitted flux is picked up by the telescope. 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_319

学习物理知识后, 我们可以用物理的视角观察周围的世界, 思考身边的事物, 并对些媒体报道的真伪做出判断。小明在浏览一些网站时, 看到了如下一些关于发射卫星的报道，其中一定不真实的消息是（） A: 发射一颗轨道与地球表面上某一纬度线 (非赤道) 为共面同心圆的地球卫星 B: 发射一颗与地球表面上某一经度线所决定的圆为共面同心圆的地球卫星 C: 发射一颗 $1 \mathrm{~h}$ 绕地球运转一周的地球卫星 D: 发射一颗每天同一时间都能通过北京上空的地球卫星

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: 学习物理知识后, 我们可以用物理的视角观察周围的世界, 思考身边的事物, 并对些媒体报道的真伪做出判断。小明在浏览一些网站时, 看到了如下一些关于发射卫星的报道，其中一定不真实的消息是（） A: 发射一颗轨道与地球表面上某一纬度线 (非赤道) 为共面同心圆的地球卫星 B: 发射一颗与地球表面上某一经度线所决定的圆为共面同心圆的地球卫星 C: 发射一颗 $1 \mathrm{~h}$ 绕地球运转一周的地球卫星 D: 发射一颗每天同一时间都能通过北京上空的地球卫星 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_151

已知一质量为 $m$ 的物体静止在北极与赤道对地面的压力差为 $\Delta N$, 假设地球是质量均匀的球体, 半径为 $R$ 。则地球的自转周期为 (设地球表面的重力加速度为 $g$ ) ( ) A: 地球的自转周期为 $T=2 \pi \sqrt{\frac{m R}{\Delta N}}$ B: 地球的自转周期为 $T=\pi \sqrt{\frac{m R}{\Delta N}}$ C: 地球同步卫星的轨道半径为 $\left(\frac{m g}{\Delta N}\right)^{\frac{1}{3}} R$ D: 地球同步卫星的轨道半径为 $2\left(\frac{m g}{\Delta N}\right)^{\frac{1}{3}} R$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 已知一质量为 $m$ 的物体静止在北极与赤道对地面的压力差为 $\Delta N$, 假设地球是质量均匀的球体, 半径为 $R$ 。则地球的自转周期为 (设地球表面的重力加速度为 $g$ ) ( ) A: 地球的自转周期为 $T=2 \pi \sqrt{\frac{m R}{\Delta N}}$ B: 地球的自转周期为 $T=\pi \sqrt{\frac{m R}{\Delta N}}$ C: 地球同步卫星的轨道半径为 $\left(\frac{m g}{\Delta N}\right)^{\frac{1}{3}} R$ D: 地球同步卫星的轨道半径为 $2\left(\frac{m g}{\Delta N}\right)^{\frac{1}{3}} R$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

text-only

Astronomy_1063

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}}$.c. In practice, the gravitational binding energy of the Earth is much lower than that of the Sun, and so the First Order would not need to drain the whole star to get enough energy to destroy the Earth. Assuming the weapon is able to channel towards it all the energy being radiated from the Sun's entire surface, how long would it take them to charge the superweapon sufficiently to do this?

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: c. In practice, the gravitational binding energy of the Earth is much lower than that of the Sun, and so the First Order would not need to drain the whole star to get enough energy to destroy the Earth. Assuming the weapon is able to channel towards it all the energy being radiated from the Sun's entire surface, how long would it take them to charge the superweapon sufficiently to do this? 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_1046

Young, Earth-like planets interact with the protoplanetary discs in which they form, and as a result migrate to different orbital radii. The aim of this question is to quantify this migration in a simple model. We shall think of protoplanetary discs as consisting of nested circular orbits of gas and dust around a central star. For a thin disc (with small vertical extent), we assign to the disc a surface density (or mass per unit area) $\Sigma$, and semi-thickness $H$, which in general vary over the disc's extent. The disc's 'aspect ratio' at radius $r$ from the central star is denoted $h=H / r$. This question is concerned with the migration of 'small' planets, such that $M_{p} / M_{\star}=q \ll h^{3}$. [figure1] Figure 6: Left: ALMA image of the young star HL Tau and its protoplanetary disk. This image of planet formation reveals multiple rings and gaps that herald the presence of emerging planets as they sweep their orbits clear of dust and gas. Credit: ALMA (NRAO/ESO/NAOJ) / C. Brogan, B. Saxton (NRAO/AUI/NSF). Right: A small planet orbits whilst embedded in a protoplanetary disc, exciting a 1-armed spiral density wave. Credit: Frdric Masset. Since the planet is assumed small $\left(q \ll h^{3}\right)$, its interaction with the gas in the disc constitutes the excitation of a spiral density wave, and redistribution of matter in the co-orbital region (that is, matter orbiting at radii $r \approx r_{p}$ ), as shown in Figure 6. The resulting non-uniform density distribution induced in the disc exerts a net gravitational force, and hence a torque on the planet, which has been estimated using analytical methods. This torque, $\Gamma$, acts to change the planet's angular momentum, and hence its orbital radius, causing it to 'migrate', via: $$ \frac{\mathrm{d} L}{\mathrm{~d} t}=\Gamma $$ It is convenient to write the torque in terms of the reference value $$ \Gamma_{0}=\left(\frac{q}{h}\right)^{2} \Sigma_{p} r_{p}^{4} \Omega_{p}^{2} $$ c. From 2-dimensional steady fluid-dynamical disc models, it is predicted that the total torque $\Gamma$ has two main contributions: from the spiral wave, the 'Lindblad torque', $\Gamma_{L}$, and from the co-orbital region, the 'Corotation torque', $\Gamma_{C}$. For a disc of uniform entropy ( $\left.\mathrm{d} s=0\right)$, and with surface density profile $\Sigma \propto r^{-\alpha}$, and pressure profile $P \propto r^{-\delta}$, Tanaka et al. (2002) and Paardekooper \& Papaloizou (2009) find these torques are given by: $$ \begin{gathered} \Gamma_{L}=(-3.20+0.86 \alpha-2.33 \delta) \Gamma_{0} \\ \Gamma_{C}=5.97(1.5-\alpha) \Gamma_{0} \end{gathered} $$ We assume the gas in the disc obeys the ideal gas law, so that: $$ \frac{P}{\Sigma T}=\text { constant }, \quad \mathrm{d} s=\text { constant } \times\left(\frac{1}{\gamma-1} \frac{\mathrm{d} T}{T}-\frac{\mathrm{d} \Sigma}{\Sigma}\right), $$ where $T$ is the absolute temperature and $\gamma$ is the adiabatic index (the ratio of the heat capacity at constant pressure to the heat capacity at constant volume). Show that for a disc of uniform entropy, $$ \Gamma=\Gamma_{L}+\Gamma_{C}=(5.76-(5.11+2.33 \gamma) \alpha) \Gamma_{0} $$ [Hint: if $\frac{\mathrm{d} y}{y}=\lambda \frac{\mathrm{d} x}{x}$, then $y \propto x^{\lambda}$.] ## Helpful equations: The moment of inertia, $I$, of a point mass $m$ moving in a circle of radius $r$ is $I=m r^{2}$. The angular momentum, $L$, of a spinning object with an angular velocity of $\Omega$ is $L=I \Omega=r \times p$, where $p$ is the linear momentum of a point particle a distance $r$ from the axis of rotation.b. Consider now a rotationally symmetric disc with surface density profile $\Sigma=\Sigma_{0}\left(r / r_{0}\right)^{-3 / 2}$, and outer radius $r_{\text {out }}=9 r_{0}$. Find the mass of the disc $M_{\text {disc }}$ in terms of $\Sigma_{0}$ and $r_{0}$ assuming its inner radius $r_{\text {in }}<<r_{0}$.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is an expression. Here is some context information for this question, which might assist you in solving it: Young, Earth-like planets interact with the protoplanetary discs in which they form, and as a result migrate to different orbital radii. The aim of this question is to quantify this migration in a simple model. We shall think of protoplanetary discs as consisting of nested circular orbits of gas and dust around a central star. For a thin disc (with small vertical extent), we assign to the disc a surface density (or mass per unit area) $\Sigma$, and semi-thickness $H$, which in general vary over the disc's extent. The disc's 'aspect ratio' at radius $r$ from the central star is denoted $h=H / r$. This question is concerned with the migration of 'small' planets, such that $M_{p} / M_{\star}=q \ll h^{3}$. [figure1] Figure 6: Left: ALMA image of the young star HL Tau and its protoplanetary disk. This image of planet formation reveals multiple rings and gaps that herald the presence of emerging planets as they sweep their orbits clear of dust and gas. Credit: ALMA (NRAO/ESO/NAOJ) / C. Brogan, B. Saxton (NRAO/AUI/NSF). Right: A small planet orbits whilst embedded in a protoplanetary disc, exciting a 1-armed spiral density wave. Credit: Frdric Masset. Since the planet is assumed small $\left(q \ll h^{3}\right)$, its interaction with the gas in the disc constitutes the excitation of a spiral density wave, and redistribution of matter in the co-orbital region (that is, matter orbiting at radii $r \approx r_{p}$ ), as shown in Figure 6. The resulting non-uniform density distribution induced in the disc exerts a net gravitational force, and hence a torque on the planet, which has been estimated using analytical methods. This torque, $\Gamma$, acts to change the planet's angular momentum, and hence its orbital radius, causing it to 'migrate', via: $$ \frac{\mathrm{d} L}{\mathrm{~d} t}=\Gamma $$ It is convenient to write the torque in terms of the reference value $$ \Gamma_{0}=\left(\frac{q}{h}\right)^{2} \Sigma_{p} r_{p}^{4} \Omega_{p}^{2} $$ c. From 2-dimensional steady fluid-dynamical disc models, it is predicted that the total torque $\Gamma$ has two main contributions: from the spiral wave, the 'Lindblad torque', $\Gamma_{L}$, and from the co-orbital region, the 'Corotation torque', $\Gamma_{C}$. For a disc of uniform entropy ( $\left.\mathrm{d} s=0\right)$, and with surface density profile $\Sigma \propto r^{-\alpha}$, and pressure profile $P \propto r^{-\delta}$, Tanaka et al. (2002) and Paardekooper \& Papaloizou (2009) find these torques are given by: $$ \begin{gathered} \Gamma_{L}=(-3.20+0.86 \alpha-2.33 \delta) \Gamma_{0} \\ \Gamma_{C}=5.97(1.5-\alpha) \Gamma_{0} \end{gathered} $$ We assume the gas in the disc obeys the ideal gas law, so that: $$ \frac{P}{\Sigma T}=\text { constant }, \quad \mathrm{d} s=\text { constant } \times\left(\frac{1}{\gamma-1} \frac{\mathrm{d} T}{T}-\frac{\mathrm{d} \Sigma}{\Sigma}\right), $$ where $T$ is the absolute temperature and $\gamma$ is the adiabatic index (the ratio of the heat capacity at constant pressure to the heat capacity at constant volume). Show that for a disc of uniform entropy, $$ \Gamma=\Gamma_{L}+\Gamma_{C}=(5.76-(5.11+2.33 \gamma) \alpha) \Gamma_{0} $$ [Hint: if $\frac{\mathrm{d} y}{y}=\lambda \frac{\mathrm{d} x}{x}$, then $y \propto x^{\lambda}$.] ## Helpful equations: The moment of inertia, $I$, of a point mass $m$ moving in a circle of radius $r$ is $I=m r^{2}$. The angular momentum, $L$, of a spinning object with an angular velocity of $\Omega$ is $L=I \Omega=r \times p$, where $p$ is the linear momentum of a point particle a distance $r$ from the axis of rotation. problem: b. Consider now a rotationally symmetric disc with surface density profile $\Sigma=\Sigma_{0}\left(r / r_{0}\right)^{-3 / 2}$, and outer radius $r_{\text {out }}=9 r_{0}$. Find the mass of the disc $M_{\text {disc }}$ in terms of $\Sigma_{0}$ and $r_{0}$ assuming its inner radius $r_{\text {in }}<<r_{0}$. 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_1077

We do not expect to find life on planets orbiting around high-mass stars because: A: High-mass stars are far too luminous B: The lifetime of a high-mass star is too short C: High-mass stars are too hot to allow for life to form D: Planets cannot have stable orbits around high-mass stars

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: We do not expect to find life on planets orbiting around high-mass stars because: A: High-mass stars are far too luminous B: The lifetime of a high-mass star is too short C: High-mass stars are too hot to allow for life to form D: Planets cannot have stable orbits around high-mass stars 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_355

设想从地球赤道平面内架设一垂直于地面延伸到太空的电梯，电梯的箱体可以将人从地面运送到地球同步轨道的空间站。已知地球表面两极处的重力加速度为 $g$, 地球自转周期为 $T$, 地球半径为 $R$, 万有引力常量为 $G$ 。求太空电梯的箱体停在距地面 $R$ 高处时, 箱体对质量为 $m$ 的乘客的作用力 $F$ 。

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: 设想从地球赤道平面内架设一垂直于地面延伸到太空的电梯，电梯的箱体可以将人从地面运送到地球同步轨道的空间站。已知地球表面两极处的重力加速度为 $g$, 地球自转周期为 $T$, 地球半径为 $R$, 万有引力常量为 $G$ 。求太空电梯的箱体停在距地面 $R$ 高处时, 箱体对质量为 $m$ 的乘客的作用力 $F$ 。 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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Astronomy_83

如图, 卫星甲、乙均绕地球做匀速圆周运动, 轨道平面互相垂直, 乙的轨道半径是试卷第 23 页，共 122 页 甲的 $\sqrt[3]{25}$ 倍, 甲做圆周运动的周期为 $T$ 。将两卫星和地心在同一直线且乙在甲正上方的位置状态称为“相遇”, 则某次“相遇”后, 甲、乙卫星再次“相遇”经历的最短时间为 ( ) [图1] A: $5 T$ B: $2.5 T$ C: $T$ D: $0.5 T$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 如图, 卫星甲、乙均绕地球做匀速圆周运动, 轨道平面互相垂直, 乙的轨道半径是试卷第 23 页，共 122 页 甲的 $\sqrt[3]{25}$ 倍, 甲做圆周运动的周期为 $T$ 。将两卫星和地心在同一直线且乙在甲正上方的位置状态称为“相遇”, 则某次“相遇”后, 甲、乙卫星再次“相遇”经历的最短时间为 ( ) [图1] A: $5 T$ B: $2.5 T$ C: $T$ D: $0.5 T$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

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multi-modal

Astronomy_648

地球质量为 $M$, 半径为 $R$, 自转周期为 $T_{0}$, 取无穷远处的引力势能为零. 质量为 $m$的卫星在绕地球无动力飞行时, 它和地球组成的系统机械能守恒, 它们之间引力势能的表达式是 $\mathrm{E}_{\mathrm{p}}=-\frac{G M m}{r}$, 其中 $r$ 是卫星与地心间的距离. 现欲将质量为 $m$ 的卫星从近地圆轨道I发射到椭圆轨道II上去, 轨道II的近地点 $A$ 和远地点 $B$ 距地心分别为 $r_{1}=R$, $r_{2}=3 R$. 若卫星在轨道II上的机械能和在 $r_{3}=2 R$ 的圆周轨道III上的机械能相同, 则 ( ) [图1] A: 卫星在近地圆轨道I上运行的周期与地球自转周期相同 B: 从轨道I发射到轨道II需要在近地的 $A$ 点一次性给它提供能量 $\frac{G M m}{4 R}$ C: 卫星在椭圆轨道上的周期为 $T_{0} \sqrt{\left(\frac{r_{2}+R}{R}\right)^{3}}$ D: 卫星在粗圆轨道II上自由运行时, 它在 $B$ 点的机械能大于在 $A$ 点的机械能

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 地球质量为 $M$, 半径为 $R$, 自转周期为 $T_{0}$, 取无穷远处的引力势能为零. 质量为 $m$的卫星在绕地球无动力飞行时, 它和地球组成的系统机械能守恒, 它们之间引力势能的表达式是 $\mathrm{E}_{\mathrm{p}}=-\frac{G M m}{r}$, 其中 $r$ 是卫星与地心间的距离. 现欲将质量为 $m$ 的卫星从近地圆轨道I发射到椭圆轨道II上去, 轨道II的近地点 $A$ 和远地点 $B$ 距地心分别为 $r_{1}=R$, $r_{2}=3 R$. 若卫星在轨道II上的机械能和在 $r_{3}=2 R$ 的圆周轨道III上的机械能相同, 则 ( ) [图1] A: 卫星在近地圆轨道I上运行的周期与地球自转周期相同 B: 从轨道I发射到轨道II需要在近地的 $A$ 点一次性给它提供能量 $\frac{G M m}{4 R}$ C: 卫星在椭圆轨道上的周期为 $T_{0} \sqrt{\left(\frac{r_{2}+R}{R}\right)^{3}}$ D: 卫星在粗圆轨道II上自由运行时, 它在 $B$ 点的机械能大于在 $A$ 点的机械能 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

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multi-modal

Astronomy_849

Let's imagine that our Universe would be filled with basketballs, each having a mass of $m_{b}=0.62$ $\mathrm{kg}$. What would be the necessary numerical density $\left(n_{b}\right)$ of basketballs in the Universe such that the mass density of the basketballs would equal the current critical density of our Universe? A: $1.5 \times 10^{-26}$ balls $/ \mathrm{m}^{3}$ B: $1.7 \times 10^{26} \mathrm{balls} / \mathrm{m}^{3}$ C: $1.5 \times 10^{-27} \mathrm{balls} / \mathrm{m}^{3}$ D: $1.7 \times 10^{27} \mathrm{balls} / \mathrm{m}^{3}$

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: Let's imagine that our Universe would be filled with basketballs, each having a mass of $m_{b}=0.62$ $\mathrm{kg}$. What would be the necessary numerical density $\left(n_{b}\right)$ of basketballs in the Universe such that the mass density of the basketballs would equal the current critical density of our Universe? A: $1.5 \times 10^{-26}$ balls $/ \mathrm{m}^{3}$ B: $1.7 \times 10^{26} \mathrm{balls} / \mathrm{m}^{3}$ C: $1.5 \times 10^{-27} \mathrm{balls} / \mathrm{m}^{3}$ D: $1.7 \times 10^{27} \mathrm{balls} / \mathrm{m}^{3}$ 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_72

2020 年 7 月 23 日, 我国首次发射火星探测器“天问一号”。地面上周期为 $2 \mathrm{~s}$ 的单摆经常被称为秒摆。假如某秒摆被“天问一号”探测器携带至火星表面后, 周期变为 $3 \mathrm{~s}$ 。已知火星半径约为地球半径的二分之一, 以下说法正确的是（） A: 若秒摆在火星表面的摆角变小，则周期也会随之变小 B: 地球质量约为火星质量的 4 倍 C: 火星的第一宇宙速度约为地球第一宇宙速度的 $\frac{\sqrt{2}}{3}$ 倍 D: “天问一号”探测器刚发射离开地球表面时, 此秒摆的周期大于 $2 \mathrm{~s}$

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: 2020 年 7 月 23 日, 我国首次发射火星探测器“天问一号”。地面上周期为 $2 \mathrm{~s}$ 的单摆经常被称为秒摆。假如某秒摆被“天问一号”探测器携带至火星表面后, 周期变为 $3 \mathrm{~s}$ 。已知火星半径约为地球半径的二分之一, 以下说法正确的是（） A: 若秒摆在火星表面的摆角变小，则周期也会随之变小 B: 地球质量约为火星质量的 4 倍 C: 火星的第一宇宙速度约为地球第一宇宙速度的 $\frac{\sqrt{2}}{3}$ 倍 D: “天问一号”探测器刚发射离开地球表面时, 此秒摆的周期大于 $2 \mathrm{~s}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

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Astronomy_942

For an observer at a latitude of $52^{\circ}$, which of these stars culminates with the highest altitude? A: Kochab $\left(\right.$ declination $\left.=74^{\circ}\right)$ B: Capella (declination $\left.=46^{\circ}\right)$ C: Vega (declination $\left.=38^{\circ}\right)$ D: Pollux (declination $=28^{\circ}$ )

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: For an observer at a latitude of $52^{\circ}$, which of these stars culminates with the highest altitude? A: Kochab $\left(\right.$ declination $\left.=74^{\circ}\right)$ B: Capella (declination $\left.=46^{\circ}\right)$ C: Vega (declination $\left.=38^{\circ}\right)$ D: Pollux (declination $=28^{\circ}$ ) 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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Astronomy_530

已知地球和火星绕太阳公转的轨道半径分别 $R_{1}$ 和 $R_{2}$, 如果把行星与太阳连线扫过的面积与其所用时间的比值定义为扫过的面积速率, 则地球和火星绕太阳公转过程中扫过的面积速率之比是（ ） A: 1 B: $\sqrt{\frac{R_{1}}{R_{2}}}$ C: $\frac{R_{2}}{R_{1}}$ D: $\sqrt{\frac{R_{2}}{R_{1}}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 已知地球和火星绕太阳公转的轨道半径分别 $R_{1}$ 和 $R_{2}$, 如果把行星与太阳连线扫过的面积与其所用时间的比值定义为扫过的面积速率, 则地球和火星绕太阳公转过程中扫过的面积速率之比是（ ） A: 1 B: $\sqrt{\frac{R_{1}}{R_{2}}}$ C: $\frac{R_{2}}{R_{1}}$ D: $\sqrt{\frac{R_{2}}{R_{1}}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

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text-only

Astronomy_503

19 世纪末, 有科学家提出了太空电梯的构想: 在赤道上建设一座直到地球同步卫星轨道的高塔, 并在塔内架设电梯。这种电梯可用于发射人造卫星, 其发射方法是将卫星通过太空电梯缓慢地提升到预定轨道高度处, 然后再启动推进装置将卫星从太空电梯发射出去, 使其直接进入预定圆轨道。已知地球质量为 $M$ 、半径为 $R$ 、自转周期为 $T$,万有引力常量为 $G$ 。 若某次通过太空电梯发射质量为 $m$ 的卫星时, 预定其轨道高度为 $h\left(h<h_{0}\right)$; 若该卫星上升到预定轨道高度时与太空电梯脱离, 脱离时卫星相对太空电梯的速度可视为零, 太空电梯把卫星运送到预定轨道高度后, 需用推进装置将卫星在预定轨道处发射进入预定轨道做匀速圆周运动, 以地心为参考系, 求推进装置需要做的功 $W$

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: 19 世纪末, 有科学家提出了太空电梯的构想: 在赤道上建设一座直到地球同步卫星轨道的高塔, 并在塔内架设电梯。这种电梯可用于发射人造卫星, 其发射方法是将卫星通过太空电梯缓慢地提升到预定轨道高度处, 然后再启动推进装置将卫星从太空电梯发射出去, 使其直接进入预定圆轨道。已知地球质量为 $M$ 、半径为 $R$ 、自转周期为 $T$,万有引力常量为 $G$ 。 若某次通过太空电梯发射质量为 $m$ 的卫星时, 预定其轨道高度为 $h\left(h<h_{0}\right)$; 若该卫星上升到预定轨道高度时与太空电梯脱离, 脱离时卫星相对太空电梯的速度可视为零, 太空电梯把卫星运送到预定轨道高度后, 需用推进装置将卫星在预定轨道处发射进入预定轨道做匀速圆周运动, 以地心为参考系, 求推进装置需要做的功 $W$ 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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text-only

Astronomy_955

Pluto has roughly the same surface area as which country? A: Australia B: China C: USA D: Russia

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: Pluto has roughly the same surface area as which country? A: Australia B: China C: USA D: Russia 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_40

卫星携带一探测器在半径为 $4 R$ 的圆轨道 $I$ 上绕地球做匀速圆周运动。在 $\mathrm{A}$ 点, 卫星上的辅助动力装置短暂工作, 将探测器沿运动方向射出 (设辅助动力装置喷出的气体质量可忽略)。若探测器恰能完全脱离地球的引力范围, 即到达距地球无限远时的速度恰好为零, 而卫星沿新的椭圆轨道II运动, 如图所示, $A 、 B$ 两点分别是其椭圆轨道II的远地点和近地点（卫星通过 $A 、 B$ 两点时的线速度大小与其距地心距离的乘积相等）。地球质量为 $M$, 探测器的质量为 $m$, 卫星的质量为 $\sqrt{2} m$, 地球半径为 $R$, 引力常量为 $G$, 已知质量分别为 $m_{1} 、 m_{2}$ 的两个质点相距为 $r$ 时, 它们之间的引力势能为 $E_{p}=-\frac{G m_{1} m_{2}}{r}$, 求: 卫星刚与探测器分离时, 卫星的线速度大小; [图1]

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 卫星携带一探测器在半径为 $4 R$ 的圆轨道 $I$ 上绕地球做匀速圆周运动。在 $\mathrm{A}$ 点, 卫星上的辅助动力装置短暂工作, 将探测器沿运动方向射出 (设辅助动力装置喷出的气体质量可忽略)。若探测器恰能完全脱离地球的引力范围, 即到达距地球无限远时的速度恰好为零, 而卫星沿新的椭圆轨道II运动, 如图所示, $A 、 B$ 两点分别是其椭圆轨道II的远地点和近地点（卫星通过 $A 、 B$ 两点时的线速度大小与其距地心距离的乘积相等）。地球质量为 $M$, 探测器的质量为 $m$, 卫星的质量为 $\sqrt{2} m$, 地球半径为 $R$, 引力常量为 $G$, 已知质量分别为 $m_{1} 、 m_{2}$ 的两个质点相距为 $r$ 时, 它们之间的引力势能为 $E_{p}=-\frac{G m_{1} m_{2}}{r}$, 求: 卫星刚与探测器分离时, 卫星的线速度大小; [图1] 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

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Astronomy

ZH

multi-modal

Astronomy_468

最近几十年, 人们对探测火星十分感兴趣, 先后发射过许多探测器。称为“火星探路者”的火星探测器曾于 1997 年登上火星。在探测器“奔向”火星的过程中, 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 $a_{1} 、 a_{2} 、 h_{0}$ 以及万有引力常量 $G$ 已知。下列判断正确的是（） [图1] A: 火星的半径为 $\frac{\sqrt{a_{2}}}{\sqrt{a_{1}}+\sqrt{a_{2}}} h_{0} $ B: 火星表面的重力加速度大小为 $a_{1}$ C: 火星的第一宇宙速度大小为 $\sqrt{\frac{a_{1} h_{0} \sqrt{a_{2}}}{\sqrt{a_{1}}+\sqrt{a_{2}}}} \quad$ D: 火星的质量大小为 $\left(\frac{\sqrt{a_{2}}}{\sqrt{a_{1}}-\sqrt{a_{2}}}\right)^{2} \frac{a_{1} h_{0}^{2}}{G}$

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: 最近几十年, 人们对探测火星十分感兴趣, 先后发射过许多探测器。称为“火星探路者”的火星探测器曾于 1997 年登上火星。在探测器“奔向”火星的过程中, 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 $a_{1} 、 a_{2} 、 h_{0}$ 以及万有引力常量 $G$ 已知。下列判断正确的是（） [图1] A: 火星的半径为 $\frac{\sqrt{a_{2}}}{\sqrt{a_{1}}+\sqrt{a_{2}}} h_{0} $ B: 火星表面的重力加速度大小为 $a_{1}$ C: 火星的第一宇宙速度大小为 $\sqrt{\frac{a_{1} h_{0} \sqrt{a_{2}}}{\sqrt{a_{1}}+\sqrt{a_{2}}}} \quad$ D: 火星的质量大小为 $\left(\frac{\sqrt{a_{2}}}{\sqrt{a_{1}}-\sqrt{a_{2}}}\right)^{2} \frac{a_{1} h_{0}^{2}}{G}$ 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]

MC

Astronomy

ZH

multi-modal

Astronomy_469

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 $\frac{\sqrt[3]{49}}{11}=\frac{1}{3}$ 。 若 $C$ 不会滑下木板 $B$, 木板 $B$ 的长度至少为多长? [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 $\frac{\sqrt[3]{49}}{11}=\frac{1}{3}$ 。 若 $C$ 不会滑下木板 $B$, 木板 $B$ 的长度至少为多长? [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m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_188

2023 年 7 月 10 日, 经国际天文学联合会小行星命名委员会批准, 中国科学院紫金山天文台发现的、国际编号为 381323 号的小行星被命名为樊锦诗星”。如图所示, “樊锦诗星”绕日运行粗圆轨道面与地球圆轨道面间的夹角为 20.11 度, 轨道半长轴为 3.18 天文单位 (日地距离为 1 天文单位), 远日点到太阳中心距离为 4.86 天文单位。下列说法正确的是（） [图1] A: “樊锦诗星”绕太阳一圈大约需要 2.15 年 B: “樊锦诗星”在远日点的速度小于地球的公转速度 C: “樊锦诗星”在远日点的加速度与地球的加速度大小之比为 $\frac{1}{4.86}$ D: “樊锦诗星”在远、近日点的速度大小之比为 $\frac{1.5}{4.86}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 2023 年 7 月 10 日, 经国际天文学联合会小行星命名委员会批准, 中国科学院紫金山天文台发现的、国际编号为 381323 号的小行星被命名为樊锦诗星”。如图所示, “樊锦诗星”绕日运行粗圆轨道面与地球圆轨道面间的夹角为 20.11 度, 轨道半长轴为 3.18 天文单位 (日地距离为 1 天文单位), 远日点到太阳中心距离为 4.86 天文单位。下列说法正确的是（） [图1] A: “樊锦诗星”绕太阳一圈大约需要 2.15 年 B: “樊锦诗星”在远日点的速度小于地球的公转速度 C: “樊锦诗星”在远日点的加速度与地球的加速度大小之比为 $\frac{1}{4.86}$ D: “樊锦诗星”在远、近日点的速度大小之比为 $\frac{1.5}{4.86}$ 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_4

2020 年 7 月 31 日, 北斗闪耀, 泽沐八方。北斗三号全球卫星导航系统（如图甲所示）建成暨开通仪式在北京举行。如图乙所示为 55 颗卫星绕地球在不同轨道上运动的 $\lg T-\lg r$ 图像, 其中 $T$ 为卫星的周期, $r$ 为卫星的轨道半径, 1 和 2 为其中的两颗卫星所对应的数据。已知引力常量为 $G$, 下列说法正确的是 ( ) [图1] 图甲 [图2] 图乙 A: 卫星 1 的周期比卫星 2 的周期小 B: 卫星 1 的周期比卫星 2 的周期大 C: 卫星 1 和 2 向心加速度大小之比为 $10^{x_{2}}: 10^{x_{1}}$ D: 卫星 1 和 2 向心加速度大小之比为 $10^{2 x_{2}}: 10^{2 x_{1}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 2020 年 7 月 31 日, 北斗闪耀, 泽沐八方。北斗三号全球卫星导航系统（如图甲所示）建成暨开通仪式在北京举行。如图乙所示为 55 颗卫星绕地球在不同轨道上运动的 $\lg T-\lg r$ 图像, 其中 $T$ 为卫星的周期, $r$ 为卫星的轨道半径, 1 和 2 为其中的两颗卫星所对应的数据。已知引力常量为 $G$, 下列说法正确的是 ( ) [图1] 图甲 [图2] 图乙 A: 卫星 1 的周期比卫星 2 的周期小 B: 卫星 1 的周期比卫星 2 的周期大 C: 卫星 1 和 2 向心加速度大小之比为 $10^{x_{2}}: 10^{x_{1}}$ D: 卫星 1 和 2 向心加速度大小之比为 $10^{2 x_{2}}: 10^{2 x_{1}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_633

2018 年 4 月 12 日, 我国遥感三十一号 01 组卫星成功发射, 用于开展电磁环境探 测. 在发射地球卫星时需要运载火箭多次点火, 以提高最终的发射速度. 某次地球近地卫星发射的过程中, 火箭喷气发动机每次喷出质量为 $m=800 \mathrm{~g}$ 的气体, 气体离开发动机时的对地速度 $v=1000 \mathrm{~m} / \mathrm{s}$, 假设火箭（含燃料在内）的总质量为 $M=600 \mathrm{~kg}$, 发动机每秒喷气 20 次, 忽略地球引力的影响, 则 A: 火箭第三次气体喷出后速度的大小约为 $4 \mathrm{~m} / \mathrm{s}$ B: 地球卫星要能成功发射, 速度大小至少达到 $11.2 \mathrm{~km} / \mathrm{s}$ C: 要使火箭能成功发射至少要喷气 500 次 D: 要使火箭能成功发射至少要持续喷气 $17 \mathrm{~s}$

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: 2018 年 4 月 12 日, 我国遥感三十一号 01 组卫星成功发射, 用于开展电磁环境探 测. 在发射地球卫星时需要运载火箭多次点火, 以提高最终的发射速度. 某次地球近地卫星发射的过程中, 火箭喷气发动机每次喷出质量为 $m=800 \mathrm{~g}$ 的气体, 气体离开发动机时的对地速度 $v=1000 \mathrm{~m} / \mathrm{s}$, 假设火箭（含燃料在内）的总质量为 $M=600 \mathrm{~kg}$, 发动机每秒喷气 20 次, 忽略地球引力的影响, 则 A: 火箭第三次气体喷出后速度的大小约为 $4 \mathrm{~m} / \mathrm{s}$ B: 地球卫星要能成功发射, 速度大小至少达到 $11.2 \mathrm{~km} / \mathrm{s}$ C: 要使火箭能成功发射至少要喷气 500 次 D: 要使火箭能成功发射至少要持续喷气 $17 \mathrm{~s}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

ZH

text-only

Astronomy_876

In the same scenario as the question above, what is the longest possible time interval that direct sunlight reaches anywhere in the bottom of the hole in a single day? A: $28 \mathrm{sec}$ B: $2 \min 8 \mathrm{sec}$ C: $2 \min 26 \mathrm{sec}$ D: $2 \min 35 \mathrm{sec}$ E: 2 min $41 \mathrm{sec}$

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 same scenario as the question above, what is the longest possible time interval that direct sunlight reaches anywhere in the bottom of the hole in a single day? A: $28 \mathrm{sec}$ B: $2 \min 8 \mathrm{sec}$ C: $2 \min 26 \mathrm{sec}$ D: $2 \min 35 \mathrm{sec}$ E: 2 min $41 \mathrm{sec}$ 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_1169

All stars lose mass during their lifetimes due to two main routes: particles escaping their surface (referred to as the stellar wind), and the mass defect of the nuclear reactions occurring in their cores. In practice, the mass loss rate can vary quite considerably during a star's lifetime, particularly once it has left the main sequence when the stellar wind can become much more substantial. Wolf-Rayet stars are massive stars near the end of their lives, presumed to be the in the stage just before a supernova, and are losing substantial amounts of mass due to very fast stellar winds. This deposits considerable energy into the surrounding interstellar medium (ISM) and can sweep up material into a thin bubble around the star, visible as a type of planetary nebula. [figure1] Figure 1: The nebula NGC 2359 around the Wolf-Rayet star WR7. The nebula is known as Thor's Helmet due to its resemblance to the helmet worn by the character from the Marvel Comics series. Credit: Star Shadows Remote Observatory and PROMPT/UNC.a. Given that, at the distance of the Earth, the proton flux from the Sun's stellar wind (which is assumed to be radiated equally in all directions) is $3.0 \times 10^{12} \mathrm{~m}^{-2} \mathrm{~s}^{-1}$, and that the luminosity of the Sun is solely due to the fusion of hydrogen to helium: i. Show that the rate at which the Sun is losing mass, $\dot{M} \equiv d M / d t$, due to its stellar wind is $\sim 10^{-14} \mathrm{M}_{\odot} \mathrm{yr}^{-1}$. [Take the mass of a proton to be $1.67 \times 10^{-27} \mathrm{~kg}$.]

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: All stars lose mass during their lifetimes due to two main routes: particles escaping their surface (referred to as the stellar wind), and the mass defect of the nuclear reactions occurring in their cores. In practice, the mass loss rate can vary quite considerably during a star's lifetime, particularly once it has left the main sequence when the stellar wind can become much more substantial. Wolf-Rayet stars are massive stars near the end of their lives, presumed to be the in the stage just before a supernova, and are losing substantial amounts of mass due to very fast stellar winds. This deposits considerable energy into the surrounding interstellar medium (ISM) and can sweep up material into a thin bubble around the star, visible as a type of planetary nebula. [figure1] Figure 1: The nebula NGC 2359 around the Wolf-Rayet star WR7. The nebula is known as Thor's Helmet due to its resemblance to the helmet worn by the character from the Marvel Comics series. Credit: Star Shadows Remote Observatory and PROMPT/UNC. problem: a. Given that, at the distance of the Earth, the proton flux from the Sun's stellar wind (which is assumed to be radiated equally in all directions) is $3.0 \times 10^{12} \mathrm{~m}^{-2} \mathrm{~s}^{-1}$, and that the luminosity of the Sun is solely due to the fusion of hydrogen to helium: i. Show that the rate at which the Sun is losing mass, $\dot{M} \equiv d M / d t$, due to its stellar wind is $\sim 10^{-14} \mathrm{M}_{\odot} \mathrm{yr}^{-1}$. [Take the mass of a proton to be $1.67 \times 10^{-27} \mathrm{~kg}$.] All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is an expression without equals signs, e.g. ANSWER=\frac{1}{2} g t^2

EX

Astronomy

EN

multi-modal

Astronomy_610

中国预计将在 2028 年实现载人登月计划, 把月球作为登上更遥远行星的一个落脚点。如图所示是“嫦娥一号奔月”的示意图, “嫦娥一号”卫星发射后经多次变轨, 进入地月转移轨道, 最终被月球引力捕获, 成为绕月卫星。关于“嫦娥一号”下列说法正确的是 [图1] A: 发射时的速度必须达到第三宇宙速度 B: 在绕地轨道中, 公转半长轴的立方与公转周期的平方之比不变 C: 在轨道I上运动时的速度小于轨道II上任意位置的速度 D: 绕月轨道II变轨到I上需点火加速

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 中国预计将在 2028 年实现载人登月计划, 把月球作为登上更遥远行星的一个落脚点。如图所示是“嫦娥一号奔月”的示意图, “嫦娥一号”卫星发射后经多次变轨, 进入地月转移轨道, 最终被月球引力捕获, 成为绕月卫星。关于“嫦娥一号”下列说法正确的是 [图1] A: 发射时的速度必须达到第三宇宙速度 B: 在绕地轨道中, 公转半长轴的立方与公转周期的平方之比不变 C: 在轨道I上运动时的速度小于轨道II上任意位置的速度 D: 绕月轨道II变轨到I上需点火加速 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_1107

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

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is 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: d. For the patched conics approach (solid lines): iii. Determine the best estimate of the duration of the real Apollo 11 translunar coast. Give your answer in hours and minutes. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is an expression without equals signs, e.g. ANSWER=\frac{1}{2} g t^2

EX

Astronomy

EN

multi-modal

Astronomy_1161

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.c. Computer models suggest the first galaxies formed around $z \sim 10-20$. One of the best ways to look for high-redshift galaxies is to try and detect the emission from the Lyman alpha (Lya) emission line at $\lambda_{\text {emit }}=121.6 \mathrm{~nm}$ as it is a relatively bright line. Some of the brightest galaxies in that initial era of galaxy formation would have an absolute magnitude of $\mathcal{M} \sim 20$. In this question, you are given that $\Omega_{0, \mathrm{~m}}=0.3, \Omega_{0, \Lambda}=0.7, \Omega_{0, \mathrm{r}}=0$ and $\mathrm{H}_{0}=70 \mathrm{~km} \mathrm{~s}^{-1} \mathrm{Mpc}^{-1}$. ii. How long after the Big Bang does this correspond to? Give you answer in years.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: 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: c. Computer models suggest the first galaxies formed around $z \sim 10-20$. One of the best ways to look for high-redshift galaxies is to try and detect the emission from the Lyman alpha (Lya) emission line at $\lambda_{\text {emit }}=121.6 \mathrm{~nm}$ as it is a relatively bright line. Some of the brightest galaxies in that initial era of galaxy formation would have an absolute magnitude of $\mathcal{M} \sim 20$. In this question, you are given that $\Omega_{0, \mathrm{~m}}=0.3, \Omega_{0, \Lambda}=0.7, \Omega_{0, \mathrm{r}}=0$ and $\mathrm{H}_{0}=70 \mathrm{~km} \mathrm{~s}^{-1} \mathrm{Mpc}^{-1}$. ii. How long after the Big Bang does this correspond to? Give you answer in years. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of years, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

multi-modal

Astronomy_1116

GW170817 was the first gravitational wave event arising from a binary neutron star merger to have been detected by the LIGO \& Virgo experiments, and careful localization of the source meant that the electromagnetic counterpart was quickly found in galaxy NGC 4993 (see Figure 2). Such a combination of two completely separate branches of astronomical observation begins a new era of 'multi-messenger astronomy'. Since the gravitational waves allow an independent measurement of the distance to the host galaxy and the light allows an independent measurement of the recessional speed, this observation allows us to determine a new, independent value of the Hubble constant $H_{0}$. Another way of measuring distances to galaxies is to use the Fundamental Plane (FP) relation, which relies on the assumption there is a fairly tight relation between radius, surface brightness, and velocity dispersion for bulge-dominated galaxies, and is widely used for galaxies like NGC 4993. It can be described by the relation $$ \log \left(\frac{D}{(1+z)^{2}}\right)=-\log R_{e}+\alpha \log \sigma-\beta \log \left\langle I_{r}\right\rangle_{e}+\gamma $$ where $D$ is the distance in Mpc, $R_{e}$ is the effective radius measured in arcseconds, $\sigma$ is the velocity dispersion in $\mathrm{km} \mathrm{s}^{-1},\left\langle I_{r}\right\rangle_{e}$ is the mean intensity inside the effective radius measured in $L_{\odot} \mathrm{pc}^{-2}$, and $\gamma$ is the distance-dependent zero point of the relation. Calibrating the zero point to the Leo I galaxy group, the constants in the FP relation become $\alpha=1.24, \beta=0.82$, and $\gamma=2.194$. [figure1] Figure 2: Left: The GW170817 signal as measured by the LIGO and Virgo gravitational wave detectors, taken from Abbott $e t$ al. (2017). The normalized amplitude (or strain) is in units of $10^{-21}$. The signal is not visible in the Virgo data due to the direction of the source with respect to the detector's antenna pattern. Right: The optical counterpart of GW170817 in host galaxy NGC 4993, taken from Hjorth et al. (2017). By measuring the amplitude (called strain, $h$ ) and the frequency of the gravitational waves, $f_{\mathrm{GW}}$, one can determine the distance to the source without having to rely on 'standard candles' like Cepheid variables or Type Ia supernovae. For two masses, $m_{1}$ and $m_{2}$, orbiting the centre of mass with separation $a$ with orbital angular velocity $\omega$, then the dimensionless strain parameter $h$ is $$ h \simeq \frac{G}{c^{4}} \frac{1}{r} \mu a^{2} \omega^{2} $$ where $r$ is the luminosity distance, $c$ is the speed of light, $\mu=m_{1} m_{2} / M_{\text {tot }}$ is the reduced mass and $M_{\text {tot }}=m_{1}+m_{2}$ is the total mass.a. Given that, at the distance of the Earth, the proton flux from the Sun's stellar wind (which is assumed to be radiated equally in all directions) is $3.0 \times 10^{12} \mathrm{~m}^{-2} \mathrm{~s}^{-1}$, and that the luminosity of the Sun is solely due to the fusion of hydrogen to helium: iii. Estimate how much the Earth-Sun distance and the Earth's orbital period will have changed after the Sun has lost mass (via both routes) for one year. Assume the orbit remains circular throughout and ignore gravitational effects from all other bodies.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: GW170817 was the first gravitational wave event arising from a binary neutron star merger to have been detected by the LIGO \& Virgo experiments, and careful localization of the source meant that the electromagnetic counterpart was quickly found in galaxy NGC 4993 (see Figure 2). Such a combination of two completely separate branches of astronomical observation begins a new era of 'multi-messenger astronomy'. Since the gravitational waves allow an independent measurement of the distance to the host galaxy and the light allows an independent measurement of the recessional speed, this observation allows us to determine a new, independent value of the Hubble constant $H_{0}$. Another way of measuring distances to galaxies is to use the Fundamental Plane (FP) relation, which relies on the assumption there is a fairly tight relation between radius, surface brightness, and velocity dispersion for bulge-dominated galaxies, and is widely used for galaxies like NGC 4993. It can be described by the relation $$ \log \left(\frac{D}{(1+z)^{2}}\right)=-\log R_{e}+\alpha \log \sigma-\beta \log \left\langle I_{r}\right\rangle_{e}+\gamma $$ where $D$ is the distance in Mpc, $R_{e}$ is the effective radius measured in arcseconds, $\sigma$ is the velocity dispersion in $\mathrm{km} \mathrm{s}^{-1},\left\langle I_{r}\right\rangle_{e}$ is the mean intensity inside the effective radius measured in $L_{\odot} \mathrm{pc}^{-2}$, and $\gamma$ is the distance-dependent zero point of the relation. Calibrating the zero point to the Leo I galaxy group, the constants in the FP relation become $\alpha=1.24, \beta=0.82$, and $\gamma=2.194$. [figure1] Figure 2: Left: The GW170817 signal as measured by the LIGO and Virgo gravitational wave detectors, taken from Abbott $e t$ al. (2017). The normalized amplitude (or strain) is in units of $10^{-21}$. The signal is not visible in the Virgo data due to the direction of the source with respect to the detector's antenna pattern. Right: The optical counterpart of GW170817 in host galaxy NGC 4993, taken from Hjorth et al. (2017). By measuring the amplitude (called strain, $h$ ) and the frequency of the gravitational waves, $f_{\mathrm{GW}}$, one can determine the distance to the source without having to rely on 'standard candles' like Cepheid variables or Type Ia supernovae. For two masses, $m_{1}$ and $m_{2}$, orbiting the centre of mass with separation $a$ with orbital angular velocity $\omega$, then the dimensionless strain parameter $h$ is $$ h \simeq \frac{G}{c^{4}} \frac{1}{r} \mu a^{2} \omega^{2} $$ where $r$ is the luminosity distance, $c$ is the speed of light, $\mu=m_{1} m_{2} / M_{\text {tot }}$ is the reduced mass and $M_{\text {tot }}=m_{1}+m_{2}$ is the total mass. problem: a. Given that, at the distance of the Earth, the proton flux from the Sun's stellar wind (which is assumed to be radiated equally in all directions) is $3.0 \times 10^{12} \mathrm{~m}^{-2} \mathrm{~s}^{-1}$, and that the luminosity of the Sun is solely due to the fusion of hydrogen to helium: iii. Estimate how much the Earth-Sun distance and the Earth's orbital period will have changed after the Sun has lost mass (via both routes) for one year. Assume the orbit remains circular throughout and ignore gravitational effects from all other bodies. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of \mathrm{~s}, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

multi-modal

Astronomy_646

2016 年 7 月 6 日, 郑州两岁女童坠 20 米深井, 消防官兵奋战 11 小时救出, 某同学为了确定矿井的深度, 查资料得知, 质量均匀分布的球壳对壳内物体的引力为零, 假定地球的密度均匀, 半径为 $R$, 矿井底部和该地表处的重力加速度大小之比为 $n$, 则该矿井的深度正确的是 ( ) A: $n R$ B: $(1-n) R$ C: $\sqrt{n} R$ D: $\left(1-\sqrt{\frac{1}{n}}\right) R$

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: 2016 年 7 月 6 日, 郑州两岁女童坠 20 米深井, 消防官兵奋战 11 小时救出, 某同学为了确定矿井的深度, 查资料得知, 质量均匀分布的球壳对壳内物体的引力为零, 假定地球的密度均匀, 半径为 $R$, 矿井底部和该地表处的重力加速度大小之比为 $n$, 则该矿井的深度正确的是 ( ) A: $n R$ B: $(1-n) R$ C: $\sqrt{n} R$ D: $\left(1-\sqrt{\frac{1}{n}}\right) R$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

text-only

Astronomy_245

人类为探索宇宙起源发射的韦伯太空望远镜运行在日地延长线上的拉格朗日 $L_{2}$ 点附近, $L_{2}$ 点的位置如图所示。在 $L_{2}$ 点的航天器受太阳和地球引力共同作用, 始终与太阳、地球保持相对静止。考虑到太阳系内其他天体的影响很小, 太阳和地球可视为以相同角速度围绕日心和地心连线中的一点 $O$ (图中未标出) 转动的双星系统。若太阳和地球的质量分别为 $M$ 和 $m$, 航天器的质量远小于太阳、地球的质量, 日心与地心的距离为 $R$, 万有引力常数为 $G, L_{2}$ 点到地心的距离记为 $r(r<<R)$, 在 $L_{2}$ 点的航天器绕 $O$点转动的角速度大小记为 $\omega$ 。下列关系式正确的是 ( ) [可能用到的近似 $\left.\frac{1}{(R+r)^{2}} \approx \frac{1}{R^{2}}\left(1-2 \frac{r}{R}\right)\right]$ [图1] A: $\omega=\left[\frac{G(M+m)}{2 R^{3}}\right]^{\frac{1}{2}}$ B: $\omega=\left[\frac{G(M+m)}{R^{3}}\right]^{\frac{1}{2}}$ C: $r=\left[\frac{3 m}{3 M+m}\right]^{\frac{1}{3}} R$ D: $r=\left[\frac{m}{3 M+m}\right]^{\frac{1}{3}} R$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 人类为探索宇宙起源发射的韦伯太空望远镜运行在日地延长线上的拉格朗日 $L_{2}$ 点附近, $L_{2}$ 点的位置如图所示。在 $L_{2}$ 点的航天器受太阳和地球引力共同作用, 始终与太阳、地球保持相对静止。考虑到太阳系内其他天体的影响很小, 太阳和地球可视为以相同角速度围绕日心和地心连线中的一点 $O$ (图中未标出) 转动的双星系统。若太阳和地球的质量分别为 $M$ 和 $m$, 航天器的质量远小于太阳、地球的质量, 日心与地心的距离为 $R$, 万有引力常数为 $G, L_{2}$ 点到地心的距离记为 $r(r<<R)$, 在 $L_{2}$ 点的航天器绕 $O$点转动的角速度大小记为 $\omega$ 。下列关系式正确的是 ( ) [可能用到的近似 $\left.\frac{1}{(R+r)^{2}} \approx \frac{1}{R^{2}}\left(1-2 \frac{r}{R}\right)\right]$ [图1] A: $\omega=\left[\frac{G(M+m)}{2 R^{3}}\right]^{\frac{1}{2}}$ B: $\omega=\left[\frac{G(M+m)}{R^{3}}\right]^{\frac{1}{2}}$ C: $r=\left[\frac{3 m}{3 M+m}\right]^{\frac{1}{3}} R$ D: $r=\left[\frac{m}{3 M+m}\right]^{\frac{1}{3}} R$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_438

2020 年 7 月 23 日 12 时 41 分, 在海南岛东北海岸中国文昌航天发射场, “天问一号”火星探测器发射成功, 一次实现火星环绕和着陆巡视探测。假设航天员登上火星后进行科学探测与实验, 航天员在火星的极地表面放置了一倾角为 $\theta$ 的斜坡, 然后从斜坡顶端以初速度 $v_{0}$ 水平抛出一个小物体，经时间 $t$ 落回到斜坡上。已知火星的半径为 $R$,自转周期为 $T_{0}$, 引力常量为 $G$, 不计阻力。则火星的（） A: 质量为 $\frac{2 v_{0} R^{2} \tan \theta}{G t}$ B: 第一宇宙速度为 $\frac{2 v_{0} R^{2} \tan \theta}{t}$ C: 密度为 $\frac{3 \pi}{G T_{0}^{2}}$ D: 同步卫星离地面高度为 $\sqrt[3]{\frac{R^{2} T_{0}^{2} v_{0} \tan \theta}{2 \pi^{2} t}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 2020 年 7 月 23 日 12 时 41 分, 在海南岛东北海岸中国文昌航天发射场, “天问一号”火星探测器发射成功, 一次实现火星环绕和着陆巡视探测。假设航天员登上火星后进行科学探测与实验, 航天员在火星的极地表面放置了一倾角为 $\theta$ 的斜坡, 然后从斜坡顶端以初速度 $v_{0}$ 水平抛出一个小物体，经时间 $t$ 落回到斜坡上。已知火星的半径为 $R$,自转周期为 $T_{0}$, 引力常量为 $G$, 不计阻力。则火星的（） A: 质量为 $\frac{2 v_{0} R^{2} \tan \theta}{G t}$ B: 第一宇宙速度为 $\frac{2 v_{0} R^{2} \tan \theta}{t}$ C: 密度为 $\frac{3 \pi}{G T_{0}^{2}}$ D: 同步卫星离地面高度为 $\sqrt[3]{\frac{R^{2} T_{0}^{2} v_{0} \tan \theta}{2 \pi^{2} t}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

text-only

Astronomy_1011

The Crab Nebula is the expanding remnant of a supernova that was observed by Chinese astronomers about 1000 years ago. Although faint in the visible, it is one of the brightest objects in the sky both in the radio and in X-ray. Figure 4 shows it as observed with the Very Large Array (a radio telescope) operating at a frequency of $3.0 \mathrm{GHz}$. The whole image has an angular size of 530 by 530 arcseconds (where there are 3600 arcseconds in a degree). [figure1] Figure 4: The Crab Nebula as seen at the radio frequency of $3.0 \mathrm{GHz}$, imaged by the Very Large Array in 2017. Credit: NRAO/AUI/NSF. For long wavelengths (such as those found in the radio), Planck's Law for black-body radiation simplifies to the Rayleigh-Jeans Law, $$ B_{\nu}=\frac{2 \nu^{2} k_{B} T}{c^{2}} $$ where $B$ is the emitted intensity at frequency $\nu$ per unit frequency per steradian (a unit of solid angle, for which there are $4 \pi$ steradians in a sphere), $k_{B}$ is the Boltzmann constant, $T$ is the temperature of the source, and $c$ is the speed of light. Typically what is measured is the emitted intensity per unit frequency, $I_{\nu}$, which is related to $B_{\nu}$ as $$ I_{\nu}=B_{\nu} \Omega $$ where $\Omega$ is the solid angle in steradians subtended by the emitting source as seen by the detector. A small angular ellipse subtends approximately $\Omega=\pi a b$ steradians, where $a$ is the semi-major axis and $b$ is the semi-minor axis of the ellipse, both specified in radians. The speed of the expansion of the nebula along its longest axis is about $1500 \mathrm{~km} \mathrm{~s}^{-1}$, although it was ejected at a higher speed during the supernova. Assuming an average deceleration of $15 \mu \mathrm{m} \mathrm{s}^{-2}$, taking the distance to the nebula as $2.0 \mathrm{kpc}$, and given the image was taken in 2017, estimate the year of the supernova.

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 Crab Nebula is the expanding remnant of a supernova that was observed by Chinese astronomers about 1000 years ago. Although faint in the visible, it is one of the brightest objects in the sky both in the radio and in X-ray. Figure 4 shows it as observed with the Very Large Array (a radio telescope) operating at a frequency of $3.0 \mathrm{GHz}$. The whole image has an angular size of 530 by 530 arcseconds (where there are 3600 arcseconds in a degree). [figure1] Figure 4: The Crab Nebula as seen at the radio frequency of $3.0 \mathrm{GHz}$, imaged by the Very Large Array in 2017. Credit: NRAO/AUI/NSF. For long wavelengths (such as those found in the radio), Planck's Law for black-body radiation simplifies to the Rayleigh-Jeans Law, $$ B_{\nu}=\frac{2 \nu^{2} k_{B} T}{c^{2}} $$ where $B$ is the emitted intensity at frequency $\nu$ per unit frequency per steradian (a unit of solid angle, for which there are $4 \pi$ steradians in a sphere), $k_{B}$ is the Boltzmann constant, $T$ is the temperature of the source, and $c$ is the speed of light. Typically what is measured is the emitted intensity per unit frequency, $I_{\nu}$, which is related to $B_{\nu}$ as $$ I_{\nu}=B_{\nu} \Omega $$ where $\Omega$ is the solid angle in steradians subtended by the emitting source as seen by the detector. A small angular ellipse subtends approximately $\Omega=\pi a b$ steradians, where $a$ is the semi-major axis and $b$ is the semi-minor axis of the ellipse, both specified in radians. The speed of the expansion of the nebula along its longest axis is about $1500 \mathrm{~km} \mathrm{~s}^{-1}$, although it was ejected at a higher speed during the supernova. Assuming an average deceleration of $15 \mu \mathrm{m} \mathrm{s}^{-2}$, taking the distance to the nebula as $2.0 \mathrm{kpc}$, and given the image was taken in 2017, estimate the year of the supernova. 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_1109

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

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: On Earth the Global Positioning System (GPS) requires a minimum of 24 satellites in orbit at any one time (there are typically more than that to allow for redundancies, with the current constellation having more than 30) so that at least 4 are visible above the horizon from anywhere on Earth (necessary for an $\mathrm{x}, \mathrm{y}, \mathrm{z}$ and time co-ordinate). This is achieved by having 6 different orbital planes, separated by $60^{\circ}$, and each orbital plane has 4 satellites. [figure1] Figure 1: The current set up of the GPS system used on Earth. Credits: Left: Peter H. Dana, University of Colorado; Right: GPS Standard Positioning Service Specification, $4^{\text {th }}$ edition The orbits are essentially circular with an eccentricity $<0.02$, an inclination of $55^{\circ}$, and an orbital period of exactly half a sidereal day (called a semi-synchronous orbit). The receiving angle of each satellite's antenna needs to be about $27.8^{\circ}$, and hence about $38 \%$ of the Earth's surface is within each satellite's footprint (see Figure 1), allowing the excellent coverage required. problem: b. How long would it take a radio signal to travel directly between a satellite and its closest neighbour in its orbital plane (assuming they're evenly spaced)? How far would a car on a motorway (with a speed of $30 \mathrm{~m} \mathrm{~s}^{-1}$ ) travel in that time? [This can be taken to be a very crude estimate of the positional accuracy of the system for that car.]. 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_727

一名宇航员登陆某星球, 对其进行探测。宇航员来到位于赤道的一平整的斜坡前,将一小球自斜坡底端正上方的 $O$ 点以初速度 $v_{0}$ 水平抛出, 如图所示。小球垂直击中了斜坡上的 $P$ 点, $P$ 点距水平地面的高度为 $h$ 。求: 若该星球的半径为 $R$, 自转周期为 $T$, 引力常量为 $G$, 则该天体的质量多大。 [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: 一名宇航员登陆某星球, 对其进行探测。宇航员来到位于赤道的一平整的斜坡前,将一小球自斜坡底端正上方的 $O$ 点以初速度 $v_{0}$ 水平抛出, 如图所示。小球垂直击中了斜坡上的 $P$ 点, $P$ 点距水平地面的高度为 $h$ 。求: 若该星球的半径为 $R$, 自转周期为 $T$, 引力常量为 $G$, 则该天体的质量多大。 [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_166

地月系统可认为是月球绕地球做匀速圆周运动如图 (a) 所示, 月球绕地球运动的周期为 $T_{1}$; 也可为地月系统是一个双星系统如图（b）所示, 在相互之间的万有引力作用下, 绕连线上的 $O$ 点做匀速圆周运动, 月球绕 $O$ 点运动的周期为 $T_{2}$ 。若地球、月球质量分别为 $M 、 m$, 两球心相距为 $r$, 万有引力常量为 $G$, 下列说法正确的是 ( ) [图1]图(a) [图2] 图(b) A: 图（a）月球绕地球运动的周期 $T_{1}$ 等于图（b）中月球绕 $O$ 点运动的周期 $T_{2}$ B: 图（a）中, 地球密度为 $\frac{3 \pi}{G T_{1}^{2}}$ C: 地月双星轨道中 $O$ 点到地心距离为 $\frac{m}{M+m} r$ D: 图（a）中, 若把部分月壤运回到地球, 最终月球绕地球做圆周运动轨道半径将变小

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 地月系统可认为是月球绕地球做匀速圆周运动如图 (a) 所示, 月球绕地球运动的周期为 $T_{1}$; 也可为地月系统是一个双星系统如图（b）所示, 在相互之间的万有引力作用下, 绕连线上的 $O$ 点做匀速圆周运动, 月球绕 $O$ 点运动的周期为 $T_{2}$ 。若地球、月球质量分别为 $M 、 m$, 两球心相距为 $r$, 万有引力常量为 $G$, 下列说法正确的是 ( ) [图1]图(a) [图2] 图(b) A: 图（a）月球绕地球运动的周期 $T_{1}$ 等于图（b）中月球绕 $O$ 点运动的周期 $T_{2}$ B: 图（a）中, 地球密度为 $\frac{3 \pi}{G T_{1}^{2}}$ C: 地月双星轨道中 $O$ 点到地心距离为 $\frac{m}{M+m} r$ D: 图（a）中, 若把部分月壤运回到地球, 最终月球绕地球做圆周运动轨道半径将变小 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_927

Which of these stars will culminate at the highest altitude as seen by an observer in Oxford (latitude $51.75^{\circ} \mathrm{N}$, longitude $1.26^{\circ} \mathrm{W}$ )? A: Aldebaran $\left(\right.$ Right ascension $=04^{\mathrm{h}} 36^{\mathrm{m}}$, declination $\left.=+16.51^{\circ}\right)$ B: Altair (Right ascension $=19^{\mathrm{h}} 51^{\mathrm{m}}$, declination $\left.=+8.87^{\circ}\right)$ C: Capella (Right ascension $=05^{\mathrm{h}} 17^{\mathrm{m}}$, declination $=+46.00^{\circ}$ ) D: Procyon $\left(\right.$ Right ascension $=07^{\mathrm{h}} 39^{\mathrm{m}}$, declination $\left.=+5.22^{\circ}\right)$

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 stars will culminate at the highest altitude as seen by an observer in Oxford (latitude $51.75^{\circ} \mathrm{N}$, longitude $1.26^{\circ} \mathrm{W}$ )? A: Aldebaran $\left(\right.$ Right ascension $=04^{\mathrm{h}} 36^{\mathrm{m}}$, declination $\left.=+16.51^{\circ}\right)$ B: Altair (Right ascension $=19^{\mathrm{h}} 51^{\mathrm{m}}$, declination $\left.=+8.87^{\circ}\right)$ C: Capella (Right ascension $=05^{\mathrm{h}} 17^{\mathrm{m}}$, declination $=+46.00^{\circ}$ ) D: Procyon $\left(\right.$ Right ascension $=07^{\mathrm{h}} 39^{\mathrm{m}}$, declination $\left.=+5.22^{\circ}\right)$ 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_718

类比是研究问题的常用方法, 科学史上很多重大发现、发明往往发端于类比。 一质量为 $m$ 的人造地球卫星绕地球做匀速圆周运动, 轨道半径为 $r$ 。将地球视为质量均匀分布的球体, 已知地球质量为 $M$, 万有引力常量为 $G$, 若质量分别为 $m_{1}$ 和 $m_{2}$ 的质点相距为 $r$ 时, 它们之间的引力势能的表达式为 $E_{\mathrm{p}}=-G \frac{m_{1} m_{2}}{r}$, 求卫星与地球组成的系统机械能。

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 类比是研究问题的常用方法, 科学史上很多重大发现、发明往往发端于类比。 一质量为 $m$ 的人造地球卫星绕地球做匀速圆周运动, 轨道半径为 $r$ 。将地球视为质量均匀分布的球体, 已知地球质量为 $M$, 万有引力常量为 $G$, 若质量分别为 $m_{1}$ 和 $m_{2}$ 的质点相距为 $r$ 时, 它们之间的引力势能的表达式为 $E_{\mathrm{p}}=-G \frac{m_{1} m_{2}}{r}$, 求卫星与地球组成的系统机械能。 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

EX

Astronomy

ZH

text-only

Astronomy_134

设想宇航员完成了对火星表面的科学考察任务, 乘坐返回舱返回围绕火星做圆周运动的轨道舱, 如图所示。为了安全, 返回舱与轨道舱对接时, 必须具有相同的速度。已知返回舱返回过程中需克服火星的引力做功 $W=m g R\left(1-\frac{R}{r}\right)$, 返同舱与人的总质量为 $m$, 火星表面的重力加速度为 $g$, 火星的半径为 $R$, 轨道舱到火星中心的距离为 $r$, 轨道舱的质量为 $M$, 不计火星表面大气对返回舱的阻力和火星自转的影响, 则下列说法正确的是 [图1] A: 该宇航员乘坐的返回舱要返回轨道舱至少需要获得能量 $W$ B: 若设无穷远处万有引力势能为零, 则地面处返回舱的引力势能为 $m g R$ C: 轨道舱的动能为 $\frac{M g R^{2}}{2 r}$ D: 若设无穷远处万有引力势能为零, 轨道舱的机械能为 $-\frac{M g R^{2}}{2 r}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 设想宇航员完成了对火星表面的科学考察任务, 乘坐返回舱返回围绕火星做圆周运动的轨道舱, 如图所示。为了安全, 返回舱与轨道舱对接时, 必须具有相同的速度。已知返回舱返回过程中需克服火星的引力做功 $W=m g R\left(1-\frac{R}{r}\right)$, 返同舱与人的总质量为 $m$, 火星表面的重力加速度为 $g$, 火星的半径为 $R$, 轨道舱到火星中心的距离为 $r$, 轨道舱的质量为 $M$, 不计火星表面大气对返回舱的阻力和火星自转的影响, 则下列说法正确的是 [图1] A: 该宇航员乘坐的返回舱要返回轨道舱至少需要获得能量 $W$ B: 若设无穷远处万有引力势能为零, 则地面处返回舱的引力势能为 $m g R$ C: 轨道舱的动能为 $\frac{M g R^{2}}{2 r}$ D: 若设无穷远处万有引力势能为零, 轨道舱的机械能为 $-\frac{M g R^{2}}{2 r}$ 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应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_584

北斗卫星导航系统是中国自主研发、独立运行的全球卫星导航系统, 北斗卫星导航系统由空间段、地面段和用户段三部分组成。空间段包括 5 颗静止轨道卫星和 30 颗非静止轨道卫星。假设一颗非静止轨道卫星 $\mathrm{a}$ 在轨道上绕行 $n$ 圈所用时间为 $t$ 。如图所示。已知地球的半径为 $R$, 地球表面处的重力加速度为 $g$, 万有引力常量为 $G$, 求: 地球的质量 $M$; [图1]

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 北斗卫星导航系统是中国自主研发、独立运行的全球卫星导航系统, 北斗卫星导航系统由空间段、地面段和用户段三部分组成。空间段包括 5 颗静止轨道卫星和 30 颗非静止轨道卫星。假设一颗非静止轨道卫星 $\mathrm{a}$ 在轨道上绕行 $n$ 圈所用时间为 $t$ 。如图所示。已知地球的半径为 $R$, 地球表面处的重力加速度为 $g$, 万有引力常量为 $G$, 求: 地球的质量 $M$; [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_1094

A day on Earth can be defined in two ways: relative to the Sun (called solar or synodic time) or relative to the background stars (called sidereal time). The mean solar day is 24 hours (within a few milliseconds), whilst the mean sidereal day is shorter at 23 hours 56 minutes 4 seconds (to the nearest second). The solar day is longer as over the course of a sidereal day the Earth has moved slightly in its orbit around the Sun and so has to rotate slightly further for the Sun to be back in the same direction (see Figure 4). [figure1] Figure 4: A solar day is defined as the time between two consecutive passages of the Sun through the meridian, corresponding to local midday (which in the Northern hemisphere is in the South), whilst a sidereal day is the time for a distant star to do the same. The difference between the two is due to the Earth having moved slightly in its orbit around the Sun. Credit: Wikipedia. The length of a year on Earth is 365.25 solar days (to 2 d.p.), however some ancient civilizations used to believe that there were once exactly 360 solar days in a year, with various myths explaining where the extra days came from. In this question you will look at how to return the Earth to this time. [Note that this question is very sensitive to the precision of the fundamental constants used, so throughout please take $G=6.674 \times 10^{-11} \mathrm{~m}^{3} \mathrm{~kg}^{-1} \mathrm{~s}^{-2}, R_{\oplus}=6371 \mathrm{~km}, M_{\oplus}=5.972 \times 10^{24} \mathrm{~kg}, M_{\odot}=$ $1.989 \times 10^{30} \mathrm{~kg}$ and $1 \mathrm{au}=1.496 \times 10^{11} \mathrm{~m}$.] ## Helpful equations: The moment of inertia, $I$, of a sphere of mass $M$ and radius $R$ is $I=\frac{2}{5} M R^{2}$. The angular momentum, $L$, of a spinning object with an angular velocity of $\omega$ is $L=I \omega=r \times p$, where $p$ is the linear momentum of a point particle a distance $r$ from the axis of rotation. The speed, $v$, of an object in an elliptical orbit of semi-major axis $a$ around an object of mass $M$ when a distance $r$ away can be calculated as $$ v^{2}=G M\left(\frac{2}{r}-\frac{1}{a}\right) $$b. Classically, two protons need to have enough energy to overcome their electrostatic repulsion in order to fuse. Calculate the value of $T_{\text {classical }}$ necessary to allow fusion to occur, given that at that point $b=1 \mathrm{fm}\left(=10^{-15} \mathrm{~m}\right)$. [You should find that it's much larger than your answer to part a.]

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: A day on Earth can be defined in two ways: relative to the Sun (called solar or synodic time) or relative to the background stars (called sidereal time). The mean solar day is 24 hours (within a few milliseconds), whilst the mean sidereal day is shorter at 23 hours 56 minutes 4 seconds (to the nearest second). The solar day is longer as over the course of a sidereal day the Earth has moved slightly in its orbit around the Sun and so has to rotate slightly further for the Sun to be back in the same direction (see Figure 4). [figure1] Figure 4: A solar day is defined as the time between two consecutive passages of the Sun through the meridian, corresponding to local midday (which in the Northern hemisphere is in the South), whilst a sidereal day is the time for a distant star to do the same. The difference between the two is due to the Earth having moved slightly in its orbit around the Sun. Credit: Wikipedia. The length of a year on Earth is 365.25 solar days (to 2 d.p.), however some ancient civilizations used to believe that there were once exactly 360 solar days in a year, with various myths explaining where the extra days came from. In this question you will look at how to return the Earth to this time. [Note that this question is very sensitive to the precision of the fundamental constants used, so throughout please take $G=6.674 \times 10^{-11} \mathrm{~m}^{3} \mathrm{~kg}^{-1} \mathrm{~s}^{-2}, R_{\oplus}=6371 \mathrm{~km}, M_{\oplus}=5.972 \times 10^{24} \mathrm{~kg}, M_{\odot}=$ $1.989 \times 10^{30} \mathrm{~kg}$ and $1 \mathrm{au}=1.496 \times 10^{11} \mathrm{~m}$.] ## Helpful equations: The moment of inertia, $I$, of a sphere of mass $M$ and radius $R$ is $I=\frac{2}{5} M R^{2}$. The angular momentum, $L$, of a spinning object with an angular velocity of $\omega$ is $L=I \omega=r \times p$, where $p$ is the linear momentum of a point particle a distance $r$ from the axis of rotation. The speed, $v$, of an object in an elliptical orbit of semi-major axis $a$ around an object of mass $M$ when a distance $r$ away can be calculated as $$ v^{2}=G M\left(\frac{2}{r}-\frac{1}{a}\right) $$ problem: b. Classically, two protons need to have enough energy to overcome their electrostatic repulsion in order to fuse. Calculate the value of $T_{\text {classical }}$ necessary to allow fusion to occur, given that at that point $b=1 \mathrm{fm}\left(=10^{-15} \mathrm{~m}\right)$. [You should find that it's much larger than your answer to part a.] 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 sidereal 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.

NV

Astronomy

EN

multi-modal

Astronomy_606

如图所示, 宇航员完成了对月球表面的科学考察任务后, 乘坐返回舱返回围绕月球做圆周运动的轨道舱。为了安全, 返回舱与轨道舱对接时, 必须具有相同的速度。已知返回舱与人的总质量为 $m$, 月球质量为 $M$, 月球的半径为 $R$, 月球表面的重力加速度为 $g$, 轨道舱到月球中心的距离为 $r$, 不计月球自转的影响。卫星绕月过程中具有的机械能由引力势能和动能组成。已知当它们相距无穷远时引力势能为零, 它们距离为 $r$ 时,引力势能为 $E_{p}=-\frac{G M m}{r}$, 则 $(\quad)$ [图1] A: 返回舱返回时，在月球表面的最大发射速度为 $v=\sqrt{g R}$ B: 返回舱在返回过程中克服引力做的功是 $W=m g R\left(1-\frac{R}{r}\right)$ C: 返回舱与轨道舱对接时应具有的动能为 $E_{k}=\frac{m g R^{2}}{2 r}$ D: 宇航员乘坐的返回舱至少需要获得 $E=m g R\left(1-\frac{R}{r}\right)$ 能量才能返回轨道舱

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, 宇航员完成了对月球表面的科学考察任务后, 乘坐返回舱返回围绕月球做圆周运动的轨道舱。为了安全, 返回舱与轨道舱对接时, 必须具有相同的速度。已知返回舱与人的总质量为 $m$, 月球质量为 $M$, 月球的半径为 $R$, 月球表面的重力加速度为 $g$, 轨道舱到月球中心的距离为 $r$, 不计月球自转的影响。卫星绕月过程中具有的机械能由引力势能和动能组成。已知当它们相距无穷远时引力势能为零, 它们距离为 $r$ 时,引力势能为 $E_{p}=-\frac{G M m}{r}$, 则 $(\quad)$ [图1] A: 返回舱返回时，在月球表面的最大发射速度为 $v=\sqrt{g R}$ B: 返回舱在返回过程中克服引力做的功是 $W=m g R\left(1-\frac{R}{r}\right)$ C: 返回舱与轨道舱对接时应具有的动能为 $E_{k}=\frac{m g R^{2}}{2 r}$ D: 宇航员乘坐的返回舱至少需要获得 $E=m g R\left(1-\frac{R}{r}\right)$ 能量才能返回轨道舱 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_755

Below, you can see the rough sketch of a telescope. What type of telescope is drawn? [figure1] A: Newtonian telescope B: Refractor telescope C: Cassegrain telescope D: Catadioptric telescope

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: Below, you can see the rough sketch of a telescope. What type of telescope is drawn? [figure1] A: Newtonian telescope B: Refractor telescope C: Cassegrain telescope D: Catadioptric telescope 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_62

中国科学院紫金山天文台于 2022 年 7 月发现两颗小行星 20220S1 和 20220N1。小行星 20220S1 预估直径约为 $230 \mathrm{~m}$, 小行星 20220N1 预估直径约为 $45 \mathrm{~m}$ 。若两小行星在同一平面内绕太阳的运动可视为匀速圆周运动 (仅考虑两小行星与太阳之间的引力),测得两小行星之间的距离 $\Delta r$ 随时间变化的关系如图所示, 已知小行星 20220S1 距太阳的距离大于小行星 20220N1 距太阳的距离。则关于小行星 20220S1 和 20220N1 的说法正确的是（） [图1] A: $20220 \mathrm{~N} 1$ 运动的周期为 $T$ B: 半径之比为 $2: 1$ C: 线速度之比为 $1: \sqrt{3}$ D: 角速度分别为 $\frac{1}{2 \sqrt{2}-1} \cdot \frac{2 \pi}{T}$ 和 $\frac{2 \sqrt{2}}{2 \sqrt{2}-1} \cdot \frac{2 \pi}{T}$

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: 中国科学院紫金山天文台于 2022 年 7 月发现两颗小行星 20220S1 和 20220N1。小行星 20220S1 预估直径约为 $230 \mathrm{~m}$, 小行星 20220N1 预估直径约为 $45 \mathrm{~m}$ 。若两小行星在同一平面内绕太阳的运动可视为匀速圆周运动 (仅考虑两小行星与太阳之间的引力),测得两小行星之间的距离 $\Delta r$ 随时间变化的关系如图所示, 已知小行星 20220S1 距太阳的距离大于小行星 20220N1 距太阳的距离。则关于小行星 20220S1 和 20220N1 的说法正确的是（） [图1] A: $20220 \mathrm{~N} 1$ 运动的周期为 $T$ B: 半径之比为 $2: 1$ C: 线速度之比为 $1: \sqrt{3}$ D: 角速度分别为 $\frac{1}{2 \sqrt{2}-1} \cdot \frac{2 \pi}{T}$ 和 $\frac{2 \sqrt{2}}{2 \sqrt{2}-1} \cdot \frac{2 \pi}{T}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

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Astronomy

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multi-modal

Astronomy_1009

A very strong argument for the existence of dark matter is: A: the existence of a supermassive black hole at the centre of most large galaxies B: the shape of the rotation curve of spiral galaxies C: the detection of neutrino emission from Type II supernovae D: that almost all galactic spectra are redshifted

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 very strong argument for the existence of dark matter is: A: the existence of a supermassive black hole at the centre of most large galaxies B: the shape of the rotation curve of spiral galaxies C: the detection of neutrino emission from Type II supernovae D: that almost all galactic spectra are redshifted 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_59

北京时间 2021 年 2 月 10 日, 中国“天问一号”探测器进入环绕火星轨道, 标志着我试卷第 52 页，共 107 页 国航天强国建设迈出坚定步伐。假设“天问一号”环绕火星的轨道半径等于某个环绕地球运动的卫星的轨道半径, 如果地球表面重力加速度是火星表面的重力加速度的 $k$ 倍, 火星的半径是地球半径的 $q$ 倍 (不考虑它们本身的自转), 火星和地球均可视为均匀球体,则下列说法正确的是 A: “天问一号”与该卫星的环绕运动周期之比为 $\sqrt{k}: q$ B: 火星与地球的密度之比为 $q: \mathrm{k}$ C: 火星与地球的第一宇宙速度之比为 $q: \sqrt{k}$ D: “天问一号”与该卫星的线速度之比为 $\sqrt{q}: k$

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 2021 年 2 月 10 日, 中国“天问一号”探测器进入环绕火星轨道, 标志着我试卷第 52 页，共 107 页 国航天强国建设迈出坚定步伐。假设“天问一号”环绕火星的轨道半径等于某个环绕地球运动的卫星的轨道半径, 如果地球表面重力加速度是火星表面的重力加速度的 $k$ 倍, 火星的半径是地球半径的 $q$ 倍 (不考虑它们本身的自转), 火星和地球均可视为均匀球体,则下列说法正确的是 A: “天问一号”与该卫星的环绕运动周期之比为 $\sqrt{k}: q$ B: 火星与地球的密度之比为 $q: \mathrm{k}$ C: 火星与地球的第一宇宙速度之比为 $q: \sqrt{k}$ D: “天问一号”与该卫星的线速度之比为 $\sqrt{q}: k$ 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_94

地球赤道上有一物体随地球的自转而做圆周运动, 所受的向心力为 $F_{1}$, 向心加速度为 $a_{1}$, 线速度为 $v_{1}$, 角速度为 $\omega_{1}$; 绕地球表面附近做圆周运动的人造卫星受的向心力为 $F_{2}$, 向心加速度为 $a_{2}$, 线速度为 $v_{2}$, 角速度为 $\omega_{2}$; 地球同步卫星所受的向心力为 $F_{3}$, 向心加速度为 $a_{3}$, 线速度为 $v_{3}$, 角速度为 $\omega_{3}$. 假设三者质量相等, 则 $(\quad)$ A: $F_{1}=F_{2}>F_{3}$ B: $a_{2}>a_{3}>a_{1}$ C: $\omega_{1}=\omega_{3}<\omega_{2}$ D: $v_{1}=v_{2}>v_{3}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 地球赤道上有一物体随地球的自转而做圆周运动, 所受的向心力为 $F_{1}$, 向心加速度为 $a_{1}$, 线速度为 $v_{1}$, 角速度为 $\omega_{1}$; 绕地球表面附近做圆周运动的人造卫星受的向心力为 $F_{2}$, 向心加速度为 $a_{2}$, 线速度为 $v_{2}$, 角速度为 $\omega_{2}$; 地球同步卫星所受的向心力为 $F_{3}$, 向心加速度为 $a_{3}$, 线速度为 $v_{3}$, 角速度为 $\omega_{3}$. 假设三者质量相等, 则 $(\quad)$ A: $F_{1}=F_{2}>F_{3}$ B: $a_{2}>a_{3}>a_{1}$ C: $\omega_{1}=\omega_{3}<\omega_{2}$ D: $v_{1}=v_{2}>v_{3}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

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Astronomy

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Astronomy_707

2020 年 7 月 23 日, “天问一号”探测器成功发射, 开启了探测火星之旅。截至 2022 年 4 月，“天问一号”已依次完成了“绕、落、巡”三大目标。假设地球近地卫星的周期与 火星近火卫星的周期比值为 $k$, 地球半径与火星半径的比值为 $n$ 。则下列说法正确的是 A: 地球质量与火星质量之比为 $n^{3}: k^{2}$ B: 地球密度与火星密度之比为 $1: k$ C: 地球第一宇宙速度与火星第一宇宙速度之比为 $\sqrt{n}: \sqrt{k}$ D: 如果地球的某一卫星与火星的某一卫星轨道半径相同, 则两卫星的加速度之比为 $n: k^{2}$

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: 2020 年 7 月 23 日, “天问一号”探测器成功发射, 开启了探测火星之旅。截至 2022 年 4 月，“天问一号”已依次完成了“绕、落、巡”三大目标。假设地球近地卫星的周期与 火星近火卫星的周期比值为 $k$, 地球半径与火星半径的比值为 $n$ 。则下列说法正确的是 A: 地球质量与火星质量之比为 $n^{3}: k^{2}$ B: 地球密度与火星密度之比为 $1: k$ C: 地球第一宇宙速度与火星第一宇宙速度之比为 $\sqrt{n}: \sqrt{k}$ D: 如果地球的某一卫星与火星的某一卫星轨道半径相同, 则两卫星的加速度之比为 $n: k^{2}$ 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_676

宇宙中存在一些离其他恒星较远的四颗星组成的四星系统。若某个四星系统中每个星体的质量均为 $m$, 半径均为 $R$, 忽略其他星体对它们的引力作用和忽略星体自转效应,则可能存在如下运动形式: 四颗星分别位于边长为 $L$ 的正方形的四个顶点上 ( $L$ 远大于 $R)$ ，在相互之间的万有引力作用下，绕某一共同的圆心做角速度相同的圆周运动。已知万有引力常量为 $G$, 则关于此四星系统, 下列说法正确的是 $(\quad)$ A: 四颗星做圆周运动的轨道半径均为 $\frac{L}{2}$ B: 四颗星表面的重力加速度均为 $G \frac{m}{R^{2}}$ C: 四颗星做圆周运动的向心力大小为 $\frac{G m^{2}}{L^{2}}(2 \sqrt{2}+1)$ D: 四颗星做圆周运动的角速度均为 $\sqrt{\frac{(4+\sqrt{2}) G m}{2 L^{3}}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 宇宙中存在一些离其他恒星较远的四颗星组成的四星系统。若某个四星系统中每个星体的质量均为 $m$, 半径均为 $R$, 忽略其他星体对它们的引力作用和忽略星体自转效应,则可能存在如下运动形式: 四颗星分别位于边长为 $L$ 的正方形的四个顶点上 ( $L$ 远大于 $R)$ ，在相互之间的万有引力作用下，绕某一共同的圆心做角速度相同的圆周运动。已知万有引力常量为 $G$, 则关于此四星系统, 下列说法正确的是 $(\quad)$ A: 四颗星做圆周运动的轨道半径均为 $\frac{L}{2}$ B: 四颗星表面的重力加速度均为 $G \frac{m}{R^{2}}$ C: 四颗星做圆周运动的向心力大小为 $\frac{G m^{2}}{L^{2}}(2 \sqrt{2}+1)$ D: 四颗星做圆周运动的角速度均为 $\sqrt{\frac{(4+\sqrt{2}) G m}{2 L^{3}}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

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Astronomy

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text-only

Astronomy_300

遥远的星球。某星球完全由不可压缩的液态水组成。星球的表面重力加速度为 $g_{0}=9.8 \mathrm{~m} / \mathrm{s}^{2}$, 半径为 $R$, 且没有自转。水的密度 $\rho=10^{3} \mathrm{~kg} / \mathrm{m}^{3}$, 万有引力常数 $G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} \cdot \mathrm{kg}^{-2}$ 。本题中可能用到如下公式: 半径为 $R$ 的球，其体积为 $V=\frac{4}{3} \pi R^{3}$, 表面积为 $S=4 \pi R^{2}$ 。星球的半径 $R$ 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数值。 问题: 遥远的星球。某星球完全由不可压缩的液态水组成。星球的表面重力加速度为 $g_{0}=9.8 \mathrm{~m} / \mathrm{s}^{2}$, 半径为 $R$, 且没有自转。水的密度 $\rho=10^{3} \mathrm{~kg} / \mathrm{m}^{3}$, 万有引力常数 $G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} \cdot \mathrm{kg}^{-2}$ 。本题中可能用到如下公式: 半径为 $R$ 的球，其体积为 $V=\frac{4}{3} \pi R^{3}$, 表面积为 $S=4 \pi R^{2}$ 。星球的半径 $R$ 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。 请记住，你的答案应以m为单位计算，但在给出最终答案时，请不要包含单位。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。

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Astronomy

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Astronomy_393

宇宙中有一星球, 其半径为 $R$, 自转周期为 $T$, 引力常量为 $G$, 该天体的质量为 $M$ 。若一宇航员来到位于赤道的一斜坡前, 将一小球自斜坡底端正上方的 $O$ 点以初速度 $v_{0}$ 水平抛出, 如图所示, 小球垂直击中了斜坡, 落点为 $P$ 点, 求 则 $P$ 点距水平地面的高度 $h$ 。 [图1]

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 宇宙中有一星球, 其半径为 $R$, 自转周期为 $T$, 引力常量为 $G$, 该天体的质量为 $M$ 。若一宇航员来到位于赤道的一斜坡前, 将一小球自斜坡底端正上方的 $O$ 点以初速度 $v_{0}$ 水平抛出, 如图所示, 小球垂直击中了斜坡, 落点为 $P$ 点, 求 则 $P$ 点距水平地面的高度 $h$ 。 [图1] 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

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Astronomy

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multi-modal

Astronomy_739

卫星围绕某行星做匀速圆周运动的轨道半径的三次方 $\left(r^{3}\right)$ 与周期的平方 $\left(T^{2}\right)$ 之间的关系如图所示。若该行星的半径 $R_{0}$ 和卫星在该行星表面运行的周期 $T_{0}$ 已知, 引力常量为 $G$, 则下列物理量中不能求出的是 ( ) [图1] A: 该卫星的线速度 B: 该卫星的动能 C: 该行星的平均密度 D: 该行星表面的重力加速度

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 卫星围绕某行星做匀速圆周运动的轨道半径的三次方 $\left(r^{3}\right)$ 与周期的平方 $\left(T^{2}\right)$ 之间的关系如图所示。若该行星的半径 $R_{0}$ 和卫星在该行星表面运行的周期 $T_{0}$ 已知, 引力常量为 $G$, 则下列物理量中不能求出的是 ( ) [图1] A: 该卫星的线速度 B: 该卫星的动能 C: 该行星的平均密度 D: 该行星表面的重力加速度 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy_65

我国天文学家通过 FAST, 在武仙座球状星团 $\mathrm{M}_{1} 3$ 中发现一个脉冲双星系统。如图所示, 假设在太空中有恒星 $A 、 B$ 双星系统绕点 $O$ 做顺时针匀速圆周运动, 运动周期为 $T_{1}$, 它们的轨道半径分别为 $R_{A} 、 R_{B}, R_{A}<R_{B}, C$ 为 $B$ 的卫星, 绕 $B$ 做逆时针匀速圆周运动, 周期为 $T_{2}$ 。忽略 $A$ 与 $C$ 之间的引力, $A$ 与 $B$ 之间的引力远大于 $C$ 与 $B$ 之间的引力。万有引力常量为 $G$, 求:恒星 $A$ 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表达式。 问题: 我国天文学家通过 FAST, 在武仙座球状星团 $\mathrm{M}_{1} 3$ 中发现一个脉冲双星系统。如图所示, 假设在太空中有恒星 $A 、 B$ 双星系统绕点 $O$ 做顺时针匀速圆周运动, 运动周期为 $T_{1}$, 它们的轨道半径分别为 $R_{A} 、 R_{B}, R_{A}<R_{B}, C$ 为 $B$ 的卫星, 绕 $B$ 做逆时针匀速圆周运动, 周期为 $T_{2}$ 。忽略 $A$ 与 $C$ 之间的引力, $A$ 与 $B$ 之间的引力远大于 $C$ 与 $B$ 之间的引力。万有引力常量为 $G$, 求:恒星 $A$ 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。 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_18

2021 年 4 月, 科学家发现, 距离地球 2764 光年的宇宙空间存在适合生命居住的双星系统, 这一发现为人类研究地外生命提供了新的思路和方向。假设构成双星系统的恒星 $a 、 b$ 距离其他天体很远, 它们都绕着二者连线上的某点做匀速圆周运动。其中恒星 $a$由于不断吸附宇宙中的尘埃而使得质量缓慢的均匀增大, 恒星 $b$ 的质量和二者之间的距离均保持不变, 两恒星均可视为质量分布均匀的球体, 则下列说法正确的是（） A: 该双星系统的角速度缓慢减小 B: 恒星 $a$ 的轨道半径缓慢增大 C: 恒星 $a$ 的动量大小缓慢增大 D: 恒星 $b$ 的线速度大小缓慢减小

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 2021 年 4 月, 科学家发现, 距离地球 2764 光年的宇宙空间存在适合生命居住的双星系统, 这一发现为人类研究地外生命提供了新的思路和方向。假设构成双星系统的恒星 $a 、 b$ 距离其他天体很远, 它们都绕着二者连线上的某点做匀速圆周运动。其中恒星 $a$由于不断吸附宇宙中的尘埃而使得质量缓慢的均匀增大, 恒星 $b$ 的质量和二者之间的距离均保持不变, 两恒星均可视为质量分布均匀的球体, 则下列说法正确的是（） A: 该双星系统的角速度缓慢减小 B: 恒星 $a$ 的轨道半径缓慢增大 C: 恒星 $a$ 的动量大小缓慢增大 D: 恒星 $b$ 的线速度大小缓慢减小 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

text-only

Astronomy_680

由教育部深空探测联合研究中心组织, 重庆大学等高校合作的“多段式多功能载运月球天梯概念研究”, 图甲是“天梯”项目海基平台效果图, 是在赤道上建造重直于水平面的“太空电梯”, 宇航员乘坐太空舱通过“太空电梯”直通地球空间站。图乙中 $r$ 为宇航员到地心的距离, $R$ 为地球半径, 曲线 $\mathrm{A}$ 为地球引力对宇航员产生的加速度大小与 $r$ 的 关系; 直线 $B$ 为宇航员由于地球自转而产生的向心加速度大小与 $r$ 的关系, 关于相对地面静止在不同高度的宇航员, 下列说法正确的是（） [图1] 甲 [图2] 乙 A: 宇航员的线速度随着 $r$ 的增大而减小 B: 图乙中 $r_{0}$ 为地球同步卫星的轨道半径 C: 宇航员在 $r=R$ 处的线速度等于第一宇宙速度 D: 宇航员感受到的“重力”随着 $r$ 的增大而增大

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: 由教育部深空探测联合研究中心组织, 重庆大学等高校合作的“多段式多功能载运月球天梯概念研究”, 图甲是“天梯”项目海基平台效果图, 是在赤道上建造重直于水平面的“太空电梯”, 宇航员乘坐太空舱通过“太空电梯”直通地球空间站。图乙中 $r$ 为宇航员到地心的距离, $R$ 为地球半径, 曲线 $\mathrm{A}$ 为地球引力对宇航员产生的加速度大小与 $r$ 的 关系; 直线 $B$ 为宇航员由于地球自转而产生的向心加速度大小与 $r$ 的关系, 关于相对地面静止在不同高度的宇航员, 下列说法正确的是（） [图1] 甲 [图2] 乙 A: 宇航员的线速度随着 $r$ 的增大而减小 B: 图乙中 $r_{0}$ 为地球同步卫星的轨道半径 C: 宇航员在 $r=R$ 处的线速度等于第一宇宙速度 D: 宇航员感受到的“重力”随着 $r$ 的增大而增大 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]

SC

Astronomy

ZH

multi-modal

Astronomy_1123

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. i. The average energy per photon is given as $\bar{E}=\frac{u}{n}=\varepsilon k_{B} T$. Find the numerical value of $\varepsilon$.

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. i. The average energy per photon is given as $\bar{E}=\frac{u}{n}=\varepsilon k_{B} T$. Find the numerical value of $\varepsilon$. 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_928

Looking up into the UK night sky in late September, which of the following bright stars rises first? A: Aldebaran $\left(\right.$ Right ascension $=04^{\mathrm{h}} 36^{\mathrm{m}}$, declination $\left.=+16.51^{\circ}\right)$ B: Rigel (Right ascension $=05^{\mathrm{h}} 15^{\mathrm{m}}$, declination $=-8.20^{\circ}$ ) C: Procyon (Right ascension $=07^{\mathrm{h}} 39^{\mathrm{m}}$, declination $=+5.23^{\circ}$ ) D: Sirius (Right ascension $=06^{\mathrm{h}} 45^{\mathrm{m}}$, declination $=-16.72^{\circ}$ )

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: Looking up into the UK night sky in late September, which of the following bright stars rises first? A: Aldebaran $\left(\right.$ Right ascension $=04^{\mathrm{h}} 36^{\mathrm{m}}$, declination $\left.=+16.51^{\circ}\right)$ B: Rigel (Right ascension $=05^{\mathrm{h}} 15^{\mathrm{m}}$, declination $=-8.20^{\circ}$ ) C: Procyon (Right ascension $=07^{\mathrm{h}} 39^{\mathrm{m}}$, declination $=+5.23^{\circ}$ ) D: Sirius (Right ascension $=06^{\mathrm{h}} 45^{\mathrm{m}}$, declination $=-16.72^{\circ}$ ) 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

EN

text-only

Astronomy_1099

A "supermoon" is a new or full moon that occurs with the Moon at or near its closest approach to Earth in a given orbit (perigee). The media commonly associates supermoons with extreme brightness and size, sometimes implying that the Moon itself will become larger and have an impact on human behaviour, but just how different is a supermoon compared to the 'normal' Moon we see each month? Lunar Data: Synodic Period Anomalistic Period Semi-major axis Orbit eccentricity $$ \begin{aligned} & =29.530589 \text { days (time between same phases e.g. full moon to full moon) } \\ & =27.554550 \text { days (time between perigees i.e. perigee to perigee) } \\ & =3.844 \times 10^{5} \mathrm{~km} \\ & =0.0549 \\ & =1738.1 \mathrm{~km} \end{aligned} $$ $$ \begin{array}{ll} \text { Radius of the Moon } & =1738.1 \mathrm{~km} \\ \text { Mass of the Moon } & =7.342 \times 10^{22} \mathrm{~kg} \end{array} $$ In this question, we will only consider a full moon that is at perigee to be a supermoon.a) Calculate how many days separate a supermoon.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: A "supermoon" is a new or full moon that occurs with the Moon at or near its closest approach to Earth in a given orbit (perigee). The media commonly associates supermoons with extreme brightness and size, sometimes implying that the Moon itself will become larger and have an impact on human behaviour, but just how different is a supermoon compared to the 'normal' Moon we see each month? Lunar Data: Synodic Period Anomalistic Period Semi-major axis Orbit eccentricity $$ \begin{aligned} & =29.530589 \text { days (time between same phases e.g. full moon to full moon) } \\ & =27.554550 \text { days (time between perigees i.e. perigee to perigee) } \\ & =3.844 \times 10^{5} \mathrm{~km} \\ & =0.0549 \\ & =1738.1 \mathrm{~km} \end{aligned} $$ $$ \begin{array}{ll} \text { Radius of the Moon } & =1738.1 \mathrm{~km} \\ \text { Mass of the Moon } & =7.342 \times 10^{22} \mathrm{~kg} \end{array} $$ In this question, we will only consider a full moon that is at perigee to be a supermoon. problem: a) Calculate how many days separate a supermoon. 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

text-only

Astronomy_1086

Young, Earth-like planets interact with the protoplanetary discs in which they form, and as a result migrate to different orbital radii. The aim of this question is to quantify this migration in a simple model. We shall think of protoplanetary discs as consisting of nested circular orbits of gas and dust around a central star. For a thin disc (with small vertical extent), we assign to the disc a surface density (or mass per unit area) $\Sigma$, and semi-thickness $H$, which in general vary over the disc's extent. The disc's 'aspect ratio' at radius $r$ from the central star is denoted $h=H / r$. This question is concerned with the migration of 'small' planets, such that $M_{p} / M_{\star}=q \ll h^{3}$. [figure1] Figure 6: Left: ALMA image of the young star HL Tau and its protoplanetary disk. This image of planet formation reveals multiple rings and gaps that herald the presence of emerging planets as they sweep their orbits clear of dust and gas. Credit: ALMA (NRAO/ESO/NAOJ) / C. Brogan, B. Saxton (NRAO/AUI/NSF). Right: A small planet orbits whilst embedded in a protoplanetary disc, exciting a 1-armed spiral density wave. Credit: Frdric Masset. Since the planet is assumed small $\left(q \ll h^{3}\right)$, its interaction with the gas in the disc constitutes the excitation of a spiral density wave, and redistribution of matter in the co-orbital region (that is, matter orbiting at radii $r \approx r_{p}$ ), as shown in Figure 6. The resulting non-uniform density distribution induced in the disc exerts a net gravitational force, and hence a torque on the planet, which has been estimated using analytical methods. This torque, $\Gamma$, acts to change the planet's angular momentum, and hence its orbital radius, causing it to 'migrate', via: $$ \frac{\mathrm{d} L}{\mathrm{~d} t}=\Gamma $$ It is convenient to write the torque in terms of the reference value $$ \Gamma_{0}=\left(\frac{q}{h}\right)^{2} \Sigma_{p} r_{p}^{4} \Omega_{p}^{2} $$ c. From 2-dimensional steady fluid-dynamical disc models, it is predicted that the total torque $\Gamma$ has two main contributions: from the spiral wave, the 'Lindblad torque', $\Gamma_{L}$, and from the co-orbital region, the 'Corotation torque', $\Gamma_{C}$. For a disc of uniform entropy ( $\left.\mathrm{d} s=0\right)$, and with surface density profile $\Sigma \propto r^{-\alpha}$, and pressure profile $P \propto r^{-\delta}$, Tanaka et al. (2002) and Paardekooper \& Papaloizou (2009) find these torques are given by: $$ \begin{gathered} \Gamma_{L}=(-3.20+0.86 \alpha-2.33 \delta) \Gamma_{0} \\ \Gamma_{C}=5.97(1.5-\alpha) \Gamma_{0} \end{gathered} $$ We assume the gas in the disc obeys the ideal gas law, so that: $$ \frac{P}{\Sigma T}=\text { constant }, \quad \mathrm{d} s=\text { constant } \times\left(\frac{1}{\gamma-1} \frac{\mathrm{d} T}{T}-\frac{\mathrm{d} \Sigma}{\Sigma}\right), $$ where $T$ is the absolute temperature and $\gamma$ is the adiabatic index (the ratio of the heat capacity at constant pressure to the heat capacity at constant volume). Show that for a disc of uniform entropy, $$ \Gamma=\Gamma_{L}+\Gamma_{C}=(5.76-(5.11+2.33 \gamma) \alpha) \Gamma_{0} $$ [Hint: if $\frac{\mathrm{d} y}{y}=\lambda \frac{\mathrm{d} x}{x}$, then $y \propto x^{\lambda}$.] ## Helpful equations: The moment of inertia, $I$, of a point mass $m$ moving in a circle of radius $r$ is $I=m r^{2}$. The angular momentum, $L$, of a spinning object with an angular velocity of $\Omega$ is $L=I \Omega=r \times p$, where $p$ is the linear momentum of a point particle a distance $r$ from the axis of rotation.d. Assume the disc model from part $\mathrm{b}$. with $\mathrm{h}=$ const. Find the time taken for the orbital radius of the planet to halve from initial radius $r_{0}$. Take $\gamma=1.4$ and give your answer in terms of the migration timescale, $t_{m}=1 / h\left(h^{3} / q\right) M * / M_{\text {disc }} \tau_{0}$, where $\tau_{0}$ is the orbital period at radius $r_{0}$.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is an expression. Here is some context information for this question, which might assist you in solving it: Young, Earth-like planets interact with the protoplanetary discs in which they form, and as a result migrate to different orbital radii. The aim of this question is to quantify this migration in a simple model. We shall think of protoplanetary discs as consisting of nested circular orbits of gas and dust around a central star. For a thin disc (with small vertical extent), we assign to the disc a surface density (or mass per unit area) $\Sigma$, and semi-thickness $H$, which in general vary over the disc's extent. The disc's 'aspect ratio' at radius $r$ from the central star is denoted $h=H / r$. This question is concerned with the migration of 'small' planets, such that $M_{p} / M_{\star}=q \ll h^{3}$. [figure1] Figure 6: Left: ALMA image of the young star HL Tau and its protoplanetary disk. This image of planet formation reveals multiple rings and gaps that herald the presence of emerging planets as they sweep their orbits clear of dust and gas. Credit: ALMA (NRAO/ESO/NAOJ) / C. Brogan, B. Saxton (NRAO/AUI/NSF). Right: A small planet orbits whilst embedded in a protoplanetary disc, exciting a 1-armed spiral density wave. Credit: Frdric Masset. Since the planet is assumed small $\left(q \ll h^{3}\right)$, its interaction with the gas in the disc constitutes the excitation of a spiral density wave, and redistribution of matter in the co-orbital region (that is, matter orbiting at radii $r \approx r_{p}$ ), as shown in Figure 6. The resulting non-uniform density distribution induced in the disc exerts a net gravitational force, and hence a torque on the planet, which has been estimated using analytical methods. This torque, $\Gamma$, acts to change the planet's angular momentum, and hence its orbital radius, causing it to 'migrate', via: $$ \frac{\mathrm{d} L}{\mathrm{~d} t}=\Gamma $$ It is convenient to write the torque in terms of the reference value $$ \Gamma_{0}=\left(\frac{q}{h}\right)^{2} \Sigma_{p} r_{p}^{4} \Omega_{p}^{2} $$ c. From 2-dimensional steady fluid-dynamical disc models, it is predicted that the total torque $\Gamma$ has two main contributions: from the spiral wave, the 'Lindblad torque', $\Gamma_{L}$, and from the co-orbital region, the 'Corotation torque', $\Gamma_{C}$. For a disc of uniform entropy ( $\left.\mathrm{d} s=0\right)$, and with surface density profile $\Sigma \propto r^{-\alpha}$, and pressure profile $P \propto r^{-\delta}$, Tanaka et al. (2002) and Paardekooper \& Papaloizou (2009) find these torques are given by: $$ \begin{gathered} \Gamma_{L}=(-3.20+0.86 \alpha-2.33 \delta) \Gamma_{0} \\ \Gamma_{C}=5.97(1.5-\alpha) \Gamma_{0} \end{gathered} $$ We assume the gas in the disc obeys the ideal gas law, so that: $$ \frac{P}{\Sigma T}=\text { constant }, \quad \mathrm{d} s=\text { constant } \times\left(\frac{1}{\gamma-1} \frac{\mathrm{d} T}{T}-\frac{\mathrm{d} \Sigma}{\Sigma}\right), $$ where $T$ is the absolute temperature and $\gamma$ is the adiabatic index (the ratio of the heat capacity at constant pressure to the heat capacity at constant volume). Show that for a disc of uniform entropy, $$ \Gamma=\Gamma_{L}+\Gamma_{C}=(5.76-(5.11+2.33 \gamma) \alpha) \Gamma_{0} $$ [Hint: if $\frac{\mathrm{d} y}{y}=\lambda \frac{\mathrm{d} x}{x}$, then $y \propto x^{\lambda}$.] ## Helpful equations: The moment of inertia, $I$, of a point mass $m$ moving in a circle of radius $r$ is $I=m r^{2}$. The angular momentum, $L$, of a spinning object with an angular velocity of $\Omega$ is $L=I \Omega=r \times p$, where $p$ is the linear momentum of a point particle a distance $r$ from the axis of rotation. problem: d. Assume the disc model from part $\mathrm{b}$. with $\mathrm{h}=$ const. Find the time taken for the orbital radius of the planet to halve from initial radius $r_{0}$. Take $\gamma=1.4$ and give your answer in terms of the migration timescale, $t_{m}=1 / h\left(h^{3} / q\right) M * / M_{\text {disc }} \tau_{0}$, where $\tau_{0}$ is the orbital period at radius $r_{0}$. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is an expression without equals signs, e.g. ANSWER=\frac{1}{2} g t^2

EX

Astronomy

EN

multi-modal

Astronomy_77

2022 年 10 月 9 日 7 时 43 分, 我国在酒泉卫星发射中心采用长征二号丁型运载火箭，成功将先进天基太阳天文台卫星“夸父一号”发射升空，卫星顺利进入预定轨道，发射任务取得了圆满成功。这颗卫星的发射, 将实现我国天基太阳探测卫星跨越式突破;我国的卫星发射技术已经占据领先地位。假设在某次卫星发射中的轨迹如图所示，设地球半径为 $R$, 地球表面的重力加速度为 $g_{0}$, 卫星在半径为 $R$ 的近地圆形轨道I上运动,到达轨道的 $A$ 点时点火变轨进入椭圆轨道II, 到达轨道的远地点 $B$ 时, 再次点火进入轨道半径为 $4 R$ 的圆形轨道III绕地球做圆周运动, 设整个过程中卫星质量保持不变, 不计空气阻力。则 ( ) [图1] A: 该卫星在轨道I、III上运行的周期之比为 $1: 8$ B: 该卫星在轨道III的运行速率小于 $\sqrt{g_{0} R}$ C: 该卫星在轨道 $I$ 上经过 $A$ 处火箭的加速度大于地球赤道上静止物体的加速度大小 D: 该卫星在轨道I上的机械能等于在轨道III上的机械能

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 2022 年 10 月 9 日 7 时 43 分, 我国在酒泉卫星发射中心采用长征二号丁型运载火箭，成功将先进天基太阳天文台卫星“夸父一号”发射升空，卫星顺利进入预定轨道，发射任务取得了圆满成功。这颗卫星的发射, 将实现我国天基太阳探测卫星跨越式突破;我国的卫星发射技术已经占据领先地位。假设在某次卫星发射中的轨迹如图所示，设地球半径为 $R$, 地球表面的重力加速度为 $g_{0}$, 卫星在半径为 $R$ 的近地圆形轨道I上运动,到达轨道的 $A$ 点时点火变轨进入椭圆轨道II, 到达轨道的远地点 $B$ 时, 再次点火进入轨道半径为 $4 R$ 的圆形轨道III绕地球做圆周运动, 设整个过程中卫星质量保持不变, 不计空气阻力。则 ( ) [图1] A: 该卫星在轨道I、III上运行的周期之比为 $1: 8$ B: 该卫星在轨道III的运行速率小于 $\sqrt{g_{0} R}$ C: 该卫星在轨道 $I$ 上经过 $A$ 处火箭的加速度大于地球赤道上静止物体的加速度大小 D: 该卫星在轨道I上的机械能等于在轨道III上的机械能 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_825

The USAAAO team successfully launched a satellite to be in the orbit within the boundary of the Solar System. The satellite took a picture of the Moon passing in front of the Sun on April 24, 2017. Estimate the distance between the Moon and the satellite. (Angular diameter of the Sun is 32') [figure1] (The copyright of this picture is reserved to NASA.) A: $\quad 5.56 \times 10^{5} \mathrm{~km}$ B: $\quad \mathbf{1 . 1 2} \times 10^{6} \mathrm{~km}$ (Answer) C: $2.24 \times 10^{6} \mathrm{~km}$ D: $1.12 \times 10^{7} \mathrm{~km}$ E: $2.24 \times 10^{7} \mathrm{~km}$

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: The USAAAO team successfully launched a satellite to be in the orbit within the boundary of the Solar System. The satellite took a picture of the Moon passing in front of the Sun on April 24, 2017. Estimate the distance between the Moon and the satellite. (Angular diameter of the Sun is 32') [figure1] (The copyright of this picture is reserved to NASA.) A: $\quad 5.56 \times 10^{5} \mathrm{~km}$ B: $\quad \mathbf{1 . 1 2} \times 10^{6} \mathrm{~km}$ (Answer) C: $2.24 \times 10^{6} \mathrm{~km}$ D: $1.12 \times 10^{7} \mathrm{~km}$ E: $2.24 \times 10^{7} \mathrm{~km}$ You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D, E].

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Astronomy

EN

multi-modal

Astronomy_293

现将火星和水星的运行轨道看成圆轨道, 如图所示。 $T_{1} 、 T_{2}$ 分别为火星和水星的公转周期, $R_{1} 、 R_{2}$ 分别为火星和水星的公转半径。则过点 $\left(\lg \frac{R_{2}}{R_{1}}, \lg \frac{T_{1}}{T_{2}}\right)$ 的直线的图像为 $(\quad)$ [图1] A: [图2] B: [图3] C: [图4] D: [图5]

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 现将火星和水星的运行轨道看成圆轨道, 如图所示。 $T_{1} 、 T_{2}$ 分别为火星和水星的公转周期, $R_{1} 、 R_{2}$ 分别为火星和水星的公转半径。则过点 $\left(\lg \frac{R_{2}}{R_{1}}, \lg \frac{T_{1}}{T_{2}}\right)$ 的直线的图像为 $(\quad)$ [图1] A: [图2] B: [图3] C: [图4] D: [图5] 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

ZH

multi-modal

Astronomy_929

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. By considering the solar wind to be radiated equally in all directions as a spherical shell, derive a formula for the rate of mass lost by the Sun, $\Delta M / \Delta t$, given the density, $\rho$, and speed, $v$, of the solar wind at a given distance $r$.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is an expression. 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. By considering the solar wind to be radiated equally in all directions as a spherical shell, derive a formula for the rate of mass lost by the Sun, $\Delta M / \Delta t$, given the density, $\rho$, and speed, $v$, of the solar wind at a given distance $r$. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is an expression without equals signs, e.g. ANSWER=\frac{1}{2} g t^2

EX

Astronomy

EN

multi-modal

Astronomy_125

如果地球的质量可以由表达式 $M=\frac{a^{b}}{G c}$ 求出, 式中 $G$ 是引力常量, $a$ 的单位是 $\mathrm{m} / \mathrm{s}$, $b$ 是 $a$ 的指数, $c$ 的单位为 $\mathrm{m} / \mathrm{s}^{2}$,下列说法正确的是 ( ) A: $a$ 是卫星绕地球运动的速度, $b=2, c$ 是地球表面的重力加速度 B: $a$ 是赤道上的物体随地球一起运动的速度, $b=2, c$ 是卫星的向心加速度 C: $a$ 是第一宇宙速度, $b=4, c$ 是地球表面的重力加速度 D: $a$ 是卫星绕地球运动的速度, $b=2, c$ 是卫星的向心加速度

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 如果地球的质量可以由表达式 $M=\frac{a^{b}}{G c}$ 求出, 式中 $G$ 是引力常量, $a$ 的单位是 $\mathrm{m} / \mathrm{s}$, $b$ 是 $a$ 的指数, $c$ 的单位为 $\mathrm{m} / \mathrm{s}^{2}$,下列说法正确的是 ( ) A: $a$ 是卫星绕地球运动的速度, $b=2, c$ 是地球表面的重力加速度 B: $a$ 是赤道上的物体随地球一起运动的速度, $b=2, c$ 是卫星的向心加速度 C: $a$ 是第一宇宙速度, $b=4, c$ 是地球表面的重力加速度 D: $a$ 是卫星绕地球运动的速度, $b=2, c$ 是卫星的向心加速度 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

ZH

text-only

Astronomy_625

如图所示, 火星车与太空舱一起绕火星做匀速圆周运动。某时刻火星车减速与太空舱分离, 并沿椭圆轨道第一次到达 $P$ 点时着陆登上火星 ( $P$ 点为粗圆长轴另一端点)。已知太空舱到火星表面的高度为火星半径的 2 倍, 火星表面的重力加速度为 $g$, 火星半径为 $R$ 。则火星车从分离到着陆所用的时间为 [图1] A: $\pi \sqrt{\frac{2 R}{g}}$ B: $2 \pi \sqrt{\frac{2 R}{g}}$ C: $\pi \sqrt{\frac{27 R}{g}}$ D: $2 \pi \sqrt{\frac{27 R}{g}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 如图所示, 火星车与太空舱一起绕火星做匀速圆周运动。某时刻火星车减速与太空舱分离, 并沿椭圆轨道第一次到达 $P$ 点时着陆登上火星 ( $P$ 点为粗圆长轴另一端点)。已知太空舱到火星表面的高度为火星半径的 2 倍, 火星表面的重力加速度为 $g$, 火星半径为 $R$ 。则火星车从分离到着陆所用的时间为 [图1] A: $\pi \sqrt{\frac{2 R}{g}}$ B: $2 \pi \sqrt{\frac{2 R}{g}}$ C: $\pi \sqrt{\frac{27 R}{g}}$ D: $2 \pi \sqrt{\frac{27 R}{g}}$ 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应为以下选项之一：[A, B, C, D]

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Astronomy

ZH

multi-modal

Astronomy_507

美国科学家在 2016 年 2 月 11 日宣布, 他们利用激光干涉引力波天文台 (LIGO) 探测到两个黑洞合并时产生的引力波，爱因斯坦在 100 年前的预测终被证实。两个黑洞在合并的过程中，某段时间内会围绕空间某一位置以相同周期做圆周运动，形成“双星”系统。设其中一个黑洞的线速度大小为 $v$, 加速度大小为 $a$, 周期为 $T$, 两黑洞的总动能为 $E$, 它们之间的距离为 $r$, 不计其他天体的影响, 两黑洞的质量不变。下列各图可能正确的是（） 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: 美国科学家在 2016 年 2 月 11 日宣布, 他们利用激光干涉引力波天文台 (LIGO) 探测到两个黑洞合并时产生的引力波，爱因斯坦在 100 年前的预测终被证实。两个黑洞在合并的过程中，某段时间内会围绕空间某一位置以相同周期做圆周运动，形成“双星”系统。设其中一个黑洞的线速度大小为 $v$, 加速度大小为 $a$, 周期为 $T$, 两黑洞的总动能为 $E$, 它们之间的距离为 $r$, 不计其他天体的影响, 两黑洞的质量不变。下列各图可能正确的是（） 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]

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Astronomy

ZH

multi-modal

Astronomy_913

Star formation begins when dense clumps within giant molecular clouds, called diffuse nebulae, begin to collapse as their gravity exceeds the pressure from the temperature of the gas. For much of the collapse the clump remains transparent and so can radiate away the energy of the collapse. Consequently the clump stays at a very constant temperature (until a protostar begins to form at the core, when the temperature does rise rapidly as the gas becomes opaque), and so this part of the collapse happens essentially in freefall. This means it can happen rather fast (by astronomical timescales!). [figure1] Figure 2: Left: Orion in visible light. Right: With radio detections of giant molecular clouds superimposed. Consider one of the very dense clumps within a nebula. Assume it is spherical with an initial radius of $r_{0}$, and begins to collapse at time $t=0$, until it has shrunk to radius $r$ by some time $t$ later. If the initial density of this clump, $\rho_{0}$, is uniform everywhere in the sphere (called a homologous collapse) then we can describe its collapse with the following formulae: $$ \theta+\frac{1}{2} \sin 2 \theta=\left(\frac{8 \pi G \rho_{0}}{3}\right)^{1 / 2} t \quad \text { where } \quad \frac{r}{r_{0}}=\cos ^{2} \theta $$ If the density of the clump is $5.0 \times 10^{-16} \mathrm{~kg} \mathrm{~m}^{-3}$, calculate a value for $t_{\mathrm{ff}}$. Give you answer in years.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. problem: Star formation begins when dense clumps within giant molecular clouds, called diffuse nebulae, begin to collapse as their gravity exceeds the pressure from the temperature of the gas. For much of the collapse the clump remains transparent and so can radiate away the energy of the collapse. Consequently the clump stays at a very constant temperature (until a protostar begins to form at the core, when the temperature does rise rapidly as the gas becomes opaque), and so this part of the collapse happens essentially in freefall. This means it can happen rather fast (by astronomical timescales!). [figure1] Figure 2: Left: Orion in visible light. Right: With radio detections of giant molecular clouds superimposed. Consider one of the very dense clumps within a nebula. Assume it is spherical with an initial radius of $r_{0}$, and begins to collapse at time $t=0$, until it has shrunk to radius $r$ by some time $t$ later. If the initial density of this clump, $\rho_{0}$, is uniform everywhere in the sphere (called a homologous collapse) then we can describe its collapse with the following formulae: $$ \theta+\frac{1}{2} \sin 2 \theta=\left(\frac{8 \pi G \rho_{0}}{3}\right)^{1 / 2} t \quad \text { where } \quad \frac{r}{r_{0}}=\cos ^{2} \theta $$ If the density of the clump is $5.0 \times 10^{-16} \mathrm{~kg} \mathrm{~m}^{-3}$, calculate a value for $t_{\mathrm{ff}}$. Give you answer in years. 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_54

宇宙中存在一些离其他恒星较远的三星系统, 通常可忽略其他星体对它们的引力作用，三星质量也相同. 现已观测到稳定的三星系统存在两种基本的构成形式: 一种是三颗星位于同一直线上，两颗星围绕中央星做圆周运动，如图甲所示; 另一种是三颗星位于等边三角形的三个顶点上, 并沿外接于等边三角形的圆形轨道运行, 如图乙所示. 设两种系统中三个星体的质量均为 $m$, 且两种系统中各星间的距离已在图甲、图乙中标出,引力常量为 $G$, 则下列说法中正确的是 ( ) [图1] 甲 [图2] 乙 A: 直线三星系统中星体做圆周运动的线速度大小为 $\sqrt{\frac{G m}{L}}$ B: 直线三星系统中星体做圆周运动的周期为 $4 \pi \sqrt{\frac{L^{3}}{5 G m}}$ C: 三角形三星系统中每颗星做圆周运动的角速度为 $2 \sqrt{\frac{L^{3}}{3 G m}}$ D: 三角形三星系统中每颗星做圆周运动的加速度大小为 $\frac{\sqrt{3} G m}{L^{2}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 宇宙中存在一些离其他恒星较远的三星系统, 通常可忽略其他星体对它们的引力作用，三星质量也相同. 现已观测到稳定的三星系统存在两种基本的构成形式: 一种是三颗星位于同一直线上，两颗星围绕中央星做圆周运动，如图甲所示; 另一种是三颗星位于等边三角形的三个顶点上, 并沿外接于等边三角形的圆形轨道运行, 如图乙所示. 设两种系统中三个星体的质量均为 $m$, 且两种系统中各星间的距离已在图甲、图乙中标出,引力常量为 $G$, 则下列说法中正确的是 ( ) [图1] 甲 [图2] 乙 A: 直线三星系统中星体做圆周运动的线速度大小为 $\sqrt{\frac{G m}{L}}$ B: 直线三星系统中星体做圆周运动的周期为 $4 \pi \sqrt{\frac{L^{3}}{5 G m}}$ C: 三角形三星系统中每颗星做圆周运动的角速度为 $2 \sqrt{\frac{L^{3}}{3 G m}}$ D: 三角形三星系统中每颗星做圆周运动的加速度大小为 $\frac{\sqrt{3} G m}{L^{2}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_1033

In September 2022, Jupiter reached opposition a few days after the autumnal equinox. Which constellation was it in at the time? [Jupiter's opposition corresponds to when it is closest in its orbit to the Earth.] A: Cancer B: Pisces C: Scorpio D: Virgo

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 September 2022, Jupiter reached opposition a few days after the autumnal equinox. Which constellation was it in at the time? [Jupiter's opposition corresponds to when it is closest in its orbit to the Earth.] A: Cancer B: Pisces C: Scorpio D: Virgo 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_748

Earth's current average surface temperature is around ... A: $5^{\circ} \mathrm{C}$ B: $10{ }^{\circ} \mathrm{C}$ C: $15^{\circ} \mathrm{C}$ D: $20^{\circ} \mathrm{C}$

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: Earth's current average surface temperature is around ... A: $5^{\circ} \mathrm{C}$ B: $10{ }^{\circ} \mathrm{C}$ C: $15^{\circ} \mathrm{C}$ D: $20^{\circ} \mathrm{C}$ 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_1189

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: i. The new luminosity of the Sun.

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: i. The new luminosity of the Sun. 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 W, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

multi-modal

Astronomy_1082

e. Calculate the projected physical separation, $r_{p}$, between the galaxy and the Voorwerp.d. The typical mass accretion rate onto an active SMBH is $\sim 2 M_{\odot} \mathrm{yr}^{-1}$ and the typical efficiency is $\eta=$ 0.1. Calculate the typical luminosity of a quasar. Compare the luminosity of the quasar with the power needed to ionize the Voorwerp.

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: e. Calculate the projected physical separation, $r_{p}$, between the galaxy and the Voorwerp. problem: d. The typical mass accretion rate onto an active SMBH is $\sim 2 M_{\odot} \mathrm{yr}^{-1}$ and the typical efficiency is $\eta=$ 0.1. Calculate the typical luminosity of a quasar. Compare the luminosity of the quasar with the power needed to ionize the Voorwerp. 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

text-only

Astronomy_586

2021 年 7 月我国成功将全球首颗民用晨昏轨道气象卫星——“云三号 05 星”送入预定圆轨道，轨道周期约为 $1.7 \mathrm{~h}$ ，被命名为“黎明星”，使我国成为国际上唯一同时拥有晨昏、上午、下午三条轨道气象卫星组网观测能力的国家, 如图所示。某时刻“黎明星”正好经过赤道上 $P$ 城市正上方，则下列说法正确的是（） [图1] A: “黎明星”做匀速圆周运动的速度大于 $7.9 \mathrm{~km} / \mathrm{s}$ B: 同步卫星的轨道半径约为“黎明星”的 10 倍 C: 该时刻后“黎明星”经过 $1.7 \mathrm{~h}$ 能经过 $P$ 城市正上方 D: 该时刻后“黎明星”经过 17 天能经过 $P$ 城市正上方

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 2021 年 7 月我国成功将全球首颗民用晨昏轨道气象卫星——“云三号 05 星”送入预定圆轨道，轨道周期约为 $1.7 \mathrm{~h}$ ，被命名为“黎明星”，使我国成为国际上唯一同时拥有晨昏、上午、下午三条轨道气象卫星组网观测能力的国家, 如图所示。某时刻“黎明星”正好经过赤道上 $P$ 城市正上方，则下列说法正确的是（） [图1] A: “黎明星”做匀速圆周运动的速度大于 $7.9 \mathrm{~km} / \mathrm{s}$ B: 同步卫星的轨道半径约为“黎明星”的 10 倍 C: 该时刻后“黎明星”经过 $1.7 \mathrm{~h}$ 能经过 $P$ 城市正上方 D: 该时刻后“黎明星”经过 17 天能经过 $P$ 城市正上方 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_939

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 $a \propto e^{H_{0} t}$ Assuming our Universe is flat, calculate the current average density of the Universe.

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 $a \propto e^{H_{0} t}$ Assuming our Universe is flat, calculate the current average density of the Universe. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of $\mathrm{~kg} \mathrm{~m}^{-3}$, 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_779

A star has the luminosity $L_{0}$. The temperature $T$ of the star doubles. How does the luminosity change? A: $2 \times L_{0}$ B: $4 \times L_{0}$ C: $8 \times L_{0}$ D: $16 \times L_{0}$

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: A star has the luminosity $L_{0}$. The temperature $T$ of the star doubles. How does the luminosity change? A: $2 \times L_{0}$ B: $4 \times L_{0}$ C: $8 \times L_{0}$ D: $16 \times L_{0}$ 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].

SC

Astronomy

EN

text-only

Astronomy_823

Now suppose that Lucas is standing still in Baia (from the previous question), and Justin is standing still on the equator. Let $P_{L 1}$ and $P_{J 1}$ be the paths of Lucas's and Justin's shadows on the summer solstice, respectively. Let $P_{L 2}$ and $P_{J 2}$ be the paths of Lucas's and Justin's shadows on the vernal equinox, respectively. Assume that the heights of Lucas and Justin are small compared to the radius of the Earth, there is no atmospheric refraction, and that the Sun is a point. Given Earth's obliquity $\varepsilon=23.44^{\circ}$, which of the following is the most specific accurate description of the shapes of each path? A: $P_{L 1}:$ Parabola, $P_{J 1}:$ Hyperbola, $P_{L 2}:$ Line, $P_{J 2}:$ Line B: $P_{L 1}:$ Parabola, $P_{J 1}:$ Hyperbola, $P_{L 2}:$ Hyperbola, $P_{J 2}:$ Line C: $P_{L 1}:$ Parabola, $P_{J 1}:$ Parabola, $P_{L 2}:$ Line, $P_{J 2}:$ Line D: $P_{L 1}:$ Hyperbola, $P_{J 1}:$ Parabola, $P_{L 2}:$ Hyperbola, $P_{J 2}:$ Line E: $P_{L 1}:$ Hyperbola, $P_{J 1}:$ Parabola, $P_{L 2}:$ Line, $P_{J 2}:$ Line

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: Now suppose that Lucas is standing still in Baia (from the previous question), and Justin is standing still on the equator. Let $P_{L 1}$ and $P_{J 1}$ be the paths of Lucas's and Justin's shadows on the summer solstice, respectively. Let $P_{L 2}$ and $P_{J 2}$ be the paths of Lucas's and Justin's shadows on the vernal equinox, respectively. Assume that the heights of Lucas and Justin are small compared to the radius of the Earth, there is no atmospheric refraction, and that the Sun is a point. Given Earth's obliquity $\varepsilon=23.44^{\circ}$, which of the following is the most specific accurate description of the shapes of each path? A: $P_{L 1}:$ Parabola, $P_{J 1}:$ Hyperbola, $P_{L 2}:$ Line, $P_{J 2}:$ Line B: $P_{L 1}:$ Parabola, $P_{J 1}:$ Hyperbola, $P_{L 2}:$ Hyperbola, $P_{J 2}:$ Line C: $P_{L 1}:$ Parabola, $P_{J 1}:$ Parabola, $P_{L 2}:$ Line, $P_{J 2}:$ Line D: $P_{L 1}:$ Hyperbola, $P_{J 1}:$ Parabola, $P_{L 2}:$ Hyperbola, $P_{J 2}:$ Line E: $P_{L 1}:$ Hyperbola, $P_{J 1}:$ Parabola, $P_{L 2}:$ Line, $P_{J 2}:$ Line 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_1191

The Event Horizon Telescope (EHT) is a project to use many widely-spaced radio telescopes as a Very Long Baseline Interferometer (VBLI) to create a virtual telescope as big as the Earth. This extraordinary size allows sufficient angular resolution to be able to image the space close to the event horizon of a super massive black hole (SMBH), and provide an opportunity to test the predictions of Einstein's theory of General Relativity (GR) in a very strong gravitational field. In April 2017 the EHT collaboration managed to co-ordinate time on all of the telescopes in the array so that they could observe the SMBH (called M87*) at the centre of the Virgo galaxy, M87, and they plan to also image the SMBH at the centre of our galaxy (called Sgr A*). [figure1] Figure 3: Left: The locations of all the telescopes used during the April 2017 observing run. The solid lines correspond to baselines used for observing M87, whilst the dashed lines were the baselines used for the calibration source. Credit: EHT Collaboration. Right: A simulated model of what the region near an SMBH could look like, modelled at much higher resolution than the EHT can achieve. The light comes from the accretion disc, but the paths of the photons are bent into a characteristic shape by the extreme gravity, leading to a 'shadow' in middle of the disc - this is what the EHT is trying to image. The left side of the image is brighter than the right side as light emitted from a substance moving towards an observer is brighter than that of one moving away. Credit: Hotaka Shiokawa. Some data about the locations of the eight telescopes in the array are given below in 3-D cartesian geocentric coordinates with $X$ pointing to the Greenwich meridian, $Y$ pointing $90^{\circ}$ away in the equatorial plane (eastern longitudes have positive $Y$ ), and positive $Z$ pointing in the direction of the North Pole. This is a left-handed coordinate system. | Facility | Location | $X(\mathrm{~m})$ | $Y(\mathrm{~m})$ | $Z(\mathrm{~m})$ | | :--- | :--- | :---: | :---: | :---: | | ALMA | Chile | 2225061.3 | -5440061.7 | -2481681.2 | | APEX | Chile | 2225039.5 | -5441197.6 | -2479303.4 | | JCMT | Hawaii, USA | -5464584.7 | -2493001.2 | 2150654.0 | | LMT | Mexico | -768715.6 | -5988507.1 | 2063354.9 | | PV | Spain | 5088967.8 | -301681.2 | 3825012.2 | | SMA | Hawaii, USA | -5464555.5 | -2492928.0 | 2150797.2 | | SMT | Arizona, USA | -1828796.2 | -5054406.8 | 3427865.2 | | SPT | Antarctica | 809.8 | -816.9 | -6359568.7 | The minimum angle, $\theta_{\min }$ (in radians) that can be resolved by a VLBI array is given by the equation $$ \theta_{\min }=\frac{\lambda_{\mathrm{obs}}}{d_{\max }}, $$ where $\lambda_{\text {obs }}$ is the observing wavelength and $d_{\max }$ is the longest straight line distance between two telescopes used (called the baseline), assumed perpendicular to the line of sight during the observation. An important length scale when discussing black holes is the gravitational radius, $r_{g}=\frac{G M}{c^{2}}$, where $G$ is the gravitational constant, $M$ is the mass of the black hole and $c$ is the speed of light. The familiar event horizon of a non-rotating black hole is called the Schwartzschild radius, $r_{S} \equiv 2 r_{g}$, however this is not what the EHT is able to observe - instead the closest it can see to a black hole is called the photon sphere, where photons orbit in the black hole in unstable circular orbits. On top of this the image of the black hole is gravitationally lensed by the black hole itself magnifying the apparent radius of the photon sphere to be between $(2 \sqrt{3+2 \sqrt{2}}) r_{g}$ and $(3 \sqrt{3}) r_{g}$, determined by spin and inclination; the latter corresponds to a perfectly spherical non-spinning black hole. The area within this lensed image will appear almost black and is the 'shadow' the EHT is looking for. [figure2] Figure 4: Four nights of data were taken for M87* during the observing window of the EHT, and whilst the diameter of the disk stayed relatively constant the location of bright spots moved, possibly indicating gas that is orbiting the black hole. Credit: EHT Collaboration. The EHT observed M87* on four separate occasions during the observing window (see Fig 4), and the team saw that some of the bright spots changed in that time, suggesting they may be associated with orbiting gas close to the black hole. The Innermost Stable Circular Orbit (ISCO) is the equivalent of the photon sphere but for particles with mass (and is also stable). The total conserved energy of a circular orbit close to a non-spinning black hole is given by $$ E=m c^{2}\left(\frac{1-\frac{2 r_{g}}{r}}{\sqrt{1-\frac{3 r_{g}}{r}}}\right) $$ and the radius of the ISCO, $r_{\mathrm{ISCO}}$, is the value of $r$ for which $E$ is minimised. We expect that most black holes are in fact spinning (since most stars are spinning) and the spin of a black hole is quantified with the spin parameter $a \equiv J / J_{\max }$ where $J$ is the angular momentum of the black hole and $J_{\max }=G M^{2} / c$ is the maximum possible angular momentum it can have. The value of $a$ varies from $-1 \leq a \leq 1$, where negative spins correspond to the black hole rotating in the opposite direction to its accretion disk, and positive spins in the same direction. If $a=1$ then $r_{\text {ISCO }}=r_{g}$, whilst if $a=-1$ then $r_{\text {ISCO }}=9 r_{g}$. The angular velocity of a particle in the ISCO is given by $$ \omega^{2}=\frac{G M}{\left(r_{\text {ISCO }}^{3 / 2}+a r_{g}^{3 / 2}\right)^{2}} $$ [figure3] Figure 5: Due to the curvature of spacetime, the real distance travelled by a particle moving from the ISCO to the photon sphere (indicated with the solid red arrow) is longer than you would get purely from subtracting the radial co-ordinates of those orbits (indicated with the dashed blue arrow), which would be valid for a flat spacetime. Relations between these distances are not to scale in this diagram. Credit: Modified from Bardeen et al. (1972). The spacetime near a black hole is curved, as described by the equations of GR. This means that the distance between two points can be substantially different to the distance you would expect if spacetime was flat. GR tells us that the proper distance travelled by a particle moving from radius $r_{1}$ to radius $r_{2}$ around a black hole of mass $M$ (with $r_{1}>r_{2}$ ) is given by $$ \Delta l=\int_{r_{2}}^{r_{1}}\left(1-\frac{2 r_{g}}{r}\right)^{-1 / 2} \mathrm{~d} r $$b. Assuming Sgr A* is a non-spinning black hole with mass $4.1 \times 10^{6} \mathrm{M}_{\odot}$ and at a distance of $8.34 \mathrm{kpc}$ : ii. Determine the angular diameter (in microarcseconds) of the lensed photon sphere of Sgr A* and hence verify that the EHT can resolve it.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is an expression. Here is some context information for this question, which might assist you in solving it: The Event Horizon Telescope (EHT) is a project to use many widely-spaced radio telescopes as a Very Long Baseline Interferometer (VBLI) to create a virtual telescope as big as the Earth. This extraordinary size allows sufficient angular resolution to be able to image the space close to the event horizon of a super massive black hole (SMBH), and provide an opportunity to test the predictions of Einstein's theory of General Relativity (GR) in a very strong gravitational field. In April 2017 the EHT collaboration managed to co-ordinate time on all of the telescopes in the array so that they could observe the SMBH (called M87*) at the centre of the Virgo galaxy, M87, and they plan to also image the SMBH at the centre of our galaxy (called Sgr A*). [figure1] Figure 3: Left: The locations of all the telescopes used during the April 2017 observing run. The solid lines correspond to baselines used for observing M87, whilst the dashed lines were the baselines used for the calibration source. Credit: EHT Collaboration. Right: A simulated model of what the region near an SMBH could look like, modelled at much higher resolution than the EHT can achieve. The light comes from the accretion disc, but the paths of the photons are bent into a characteristic shape by the extreme gravity, leading to a 'shadow' in middle of the disc - this is what the EHT is trying to image. The left side of the image is brighter than the right side as light emitted from a substance moving towards an observer is brighter than that of one moving away. Credit: Hotaka Shiokawa. Some data about the locations of the eight telescopes in the array are given below in 3-D cartesian geocentric coordinates with $X$ pointing to the Greenwich meridian, $Y$ pointing $90^{\circ}$ away in the equatorial plane (eastern longitudes have positive $Y$ ), and positive $Z$ pointing in the direction of the North Pole. This is a left-handed coordinate system. | Facility | Location | $X(\mathrm{~m})$ | $Y(\mathrm{~m})$ | $Z(\mathrm{~m})$ | | :--- | :--- | :---: | :---: | :---: | | ALMA | Chile | 2225061.3 | -5440061.7 | -2481681.2 | | APEX | Chile | 2225039.5 | -5441197.6 | -2479303.4 | | JCMT | Hawaii, USA | -5464584.7 | -2493001.2 | 2150654.0 | | LMT | Mexico | -768715.6 | -5988507.1 | 2063354.9 | | PV | Spain | 5088967.8 | -301681.2 | 3825012.2 | | SMA | Hawaii, USA | -5464555.5 | -2492928.0 | 2150797.2 | | SMT | Arizona, USA | -1828796.2 | -5054406.8 | 3427865.2 | | SPT | Antarctica | 809.8 | -816.9 | -6359568.7 | The minimum angle, $\theta_{\min }$ (in radians) that can be resolved by a VLBI array is given by the equation $$ \theta_{\min }=\frac{\lambda_{\mathrm{obs}}}{d_{\max }}, $$ where $\lambda_{\text {obs }}$ is the observing wavelength and $d_{\max }$ is the longest straight line distance between two telescopes used (called the baseline), assumed perpendicular to the line of sight during the observation. An important length scale when discussing black holes is the gravitational radius, $r_{g}=\frac{G M}{c^{2}}$, where $G$ is the gravitational constant, $M$ is the mass of the black hole and $c$ is the speed of light. The familiar event horizon of a non-rotating black hole is called the Schwartzschild radius, $r_{S} \equiv 2 r_{g}$, however this is not what the EHT is able to observe - instead the closest it can see to a black hole is called the photon sphere, where photons orbit in the black hole in unstable circular orbits. On top of this the image of the black hole is gravitationally lensed by the black hole itself magnifying the apparent radius of the photon sphere to be between $(2 \sqrt{3+2 \sqrt{2}}) r_{g}$ and $(3 \sqrt{3}) r_{g}$, determined by spin and inclination; the latter corresponds to a perfectly spherical non-spinning black hole. The area within this lensed image will appear almost black and is the 'shadow' the EHT is looking for. [figure2] Figure 4: Four nights of data were taken for M87* during the observing window of the EHT, and whilst the diameter of the disk stayed relatively constant the location of bright spots moved, possibly indicating gas that is orbiting the black hole. Credit: EHT Collaboration. The EHT observed M87* on four separate occasions during the observing window (see Fig 4), and the team saw that some of the bright spots changed in that time, suggesting they may be associated with orbiting gas close to the black hole. The Innermost Stable Circular Orbit (ISCO) is the equivalent of the photon sphere but for particles with mass (and is also stable). The total conserved energy of a circular orbit close to a non-spinning black hole is given by $$ E=m c^{2}\left(\frac{1-\frac{2 r_{g}}{r}}{\sqrt{1-\frac{3 r_{g}}{r}}}\right) $$ and the radius of the ISCO, $r_{\mathrm{ISCO}}$, is the value of $r$ for which $E$ is minimised. We expect that most black holes are in fact spinning (since most stars are spinning) and the spin of a black hole is quantified with the spin parameter $a \equiv J / J_{\max }$ where $J$ is the angular momentum of the black hole and $J_{\max }=G M^{2} / c$ is the maximum possible angular momentum it can have. The value of $a$ varies from $-1 \leq a \leq 1$, where negative spins correspond to the black hole rotating in the opposite direction to its accretion disk, and positive spins in the same direction. If $a=1$ then $r_{\text {ISCO }}=r_{g}$, whilst if $a=-1$ then $r_{\text {ISCO }}=9 r_{g}$. The angular velocity of a particle in the ISCO is given by $$ \omega^{2}=\frac{G M}{\left(r_{\text {ISCO }}^{3 / 2}+a r_{g}^{3 / 2}\right)^{2}} $$ [figure3] Figure 5: Due to the curvature of spacetime, the real distance travelled by a particle moving from the ISCO to the photon sphere (indicated with the solid red arrow) is longer than you would get purely from subtracting the radial co-ordinates of those orbits (indicated with the dashed blue arrow), which would be valid for a flat spacetime. Relations between these distances are not to scale in this diagram. Credit: Modified from Bardeen et al. (1972). The spacetime near a black hole is curved, as described by the equations of GR. This means that the distance between two points can be substantially different to the distance you would expect if spacetime was flat. GR tells us that the proper distance travelled by a particle moving from radius $r_{1}$ to radius $r_{2}$ around a black hole of mass $M$ (with $r_{1}>r_{2}$ ) is given by $$ \Delta l=\int_{r_{2}}^{r_{1}}\left(1-\frac{2 r_{g}}{r}\right)^{-1 / 2} \mathrm{~d} r $$ problem: b. Assuming Sgr A* is a non-spinning black hole with mass $4.1 \times 10^{6} \mathrm{M}_{\odot}$ and at a distance of $8.34 \mathrm{kpc}$ : ii. Determine the angular diameter (in microarcseconds) of the lensed photon sphere of Sgr A* and hence verify that the EHT can resolve it. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is an expression without equals signs, e.g. ANSWER=\frac{1}{2} g t^2

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Astronomy

EN

multi-modal

Astronomy_810

Kerbyn is a small rocky planet in a circular orbit around a $0.2 M$ star with a semimajor axis of $0.1 A U$. Kerbyn has an axial tilt of $\epsilon=42^{\circ}$ and a sidereal rotation period of $05^{h} 59^{m} 9.4^{s}$. On the vernal equinox, what is the length of the apparent solar day on Kerbyn? The apparent solar day is defined as the interval between successive crossings of the meridian by the sun. A: $05^{h} 55^{m} 39.3^{s}$ B: $05^{h} 57^{m} 15.2^{s}$ C: $06^{h} 00^{m} 00.0^{s}$ D: $06^{h} 01^{m} 45.1^{s}$ E: $06^{h} 02^{m} 39.5^{s}$

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: Kerbyn is a small rocky planet in a circular orbit around a $0.2 M$ star with a semimajor axis of $0.1 A U$. Kerbyn has an axial tilt of $\epsilon=42^{\circ}$ and a sidereal rotation period of $05^{h} 59^{m} 9.4^{s}$. On the vernal equinox, what is the length of the apparent solar day on Kerbyn? The apparent solar day is defined as the interval between successive crossings of the meridian by the sun. A: $05^{h} 55^{m} 39.3^{s}$ B: $05^{h} 57^{m} 15.2^{s}$ C: $06^{h} 00^{m} 00.0^{s}$ D: $06^{h} 01^{m} 45.1^{s}$ E: $06^{h} 02^{m} 39.5^{s}$ You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D, E].

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Astronomy

EN

text-only

Astronomy_617

宇航员抵达一半径为 $R$ 的星球后，做了如下的实验 取一根细绳穿过光滑的细直管,细绳的一端拴一质量为 $m$ 的砝码, 另一端连接在一固定的拉力传感器上, 手捏细直管抢动砝码, 使它在坚直平面内做圆周运动. 若该星球表面没有空气, 不计阻力, 停止抢动细直管, 砝码可继续在同一坚直平面内做完整的圆周运动, 如图所示, 此时拉力传感器显示砝码运动到最低点与最高点两位置时读数差的绝对值为 $\Delta F$. 已知万有引力常量为 $G$, 根据题中提供的条件和测量结果, 可知( ) [图1] A: 该星球表面的重力加速度为 $\frac{\Delta F}{2 m}$ B: 该星球表面的重力加速度为 $\frac{\Delta F}{6 m}$ C: 该星球的质量为 $\frac{\Delta F R^{2}}{6 G m}$ D: 该星球的质量为 $\frac{\Delta F R^{2}}{3 G m}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 宇航员抵达一半径为 $R$ 的星球后，做了如下的实验 取一根细绳穿过光滑的细直管,细绳的一端拴一质量为 $m$ 的砝码, 另一端连接在一固定的拉力传感器上, 手捏细直管抢动砝码, 使它在坚直平面内做圆周运动. 若该星球表面没有空气, 不计阻力, 停止抢动细直管, 砝码可继续在同一坚直平面内做完整的圆周运动, 如图所示, 此时拉力传感器显示砝码运动到最低点与最高点两位置时读数差的绝对值为 $\Delta F$. 已知万有引力常量为 $G$, 根据题中提供的条件和测量结果, 可知( ) [图1] A: 该星球表面的重力加速度为 $\frac{\Delta F}{2 m}$ B: 该星球表面的重力加速度为 $\frac{\Delta F}{6 m}$ C: 该星球的质量为 $\frac{\Delta F R^{2}}{6 G m}$ D: 该星球的质量为 $\frac{\Delta F R^{2}}{3 G m}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_756

The surface temperature (in ${ }^{\circ} \mathrm{C}$ ) of the Sun is close to ... A: $5200{ }^{\circ} \mathrm{C}$ B: $5500^{\circ} \mathrm{C}$ C: $5800{ }^{\circ} \mathrm{C}$ D: $6000^{\circ} \mathrm{C}$

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 surface temperature (in ${ }^{\circ} \mathrm{C}$ ) of the Sun is close to ... A: $5200{ }^{\circ} \mathrm{C}$ B: $5500^{\circ} \mathrm{C}$ C: $5800{ }^{\circ} \mathrm{C}$ D: $6000^{\circ} \mathrm{C}$ 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_210

太空中存在一些离其他恒星很远的、由两颗星体组成的双星系统, 可忽略其他星体对它们的引力作用。如果将某双星系统简化为理想的圆周运动模型, 如图所示, 两星球在相互的万有引力作用下, 绕 $\mathrm{O}$ 点做匀速圆周运动。由于双星间的距离减小，则 $(\quad)$ [图1] A: 两星的运动角速度均逐渐减小 B: 两星的运动周期均逐渐减小 C: 两星的向心加速度均逐渐减小 D: 两星的运动线速度均逐渐减小

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 太空中存在一些离其他恒星很远的、由两颗星体组成的双星系统, 可忽略其他星体对它们的引力作用。如果将某双星系统简化为理想的圆周运动模型, 如图所示, 两星球在相互的万有引力作用下, 绕 $\mathrm{O}$ 点做匀速圆周运动。由于双星间的距离减小，则 $(\quad)$ [图1] A: 两星的运动角速度均逐渐减小 B: 两星的运动周期均逐渐减小 C: 两星的向心加速度均逐渐减小 D: 两星的运动线速度均逐渐减小 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

ZH

multi-modal

Astronomy_669

压强表示单位面积上压力的大小, 是物理学中的重要概念。 单个粒子碰撞在某一平面上会产生一个短暂的作用力, 而大量粒子持续碰撞会产生一个持续的作用力。一束均匀粒子流持续碰撞一平面, 设该束粒子流中每个粒子的质量均为 $m$ 、速度大小均为 $v$, 方向都与该平面垂直, 单位体积内的粒子数为 $n$, 粒子与该平面碰撞后均不反弹，忽略空气阻力，不考虑粒子所受重力以及粒子间的相互作用。求粒子流对该平面所产生的压强 $p$ 。

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: 压强表示单位面积上压力的大小, 是物理学中的重要概念。 单个粒子碰撞在某一平面上会产生一个短暂的作用力, 而大量粒子持续碰撞会产生一个持续的作用力。一束均匀粒子流持续碰撞一平面, 设该束粒子流中每个粒子的质量均为 $m$ 、速度大小均为 $v$, 方向都与该平面垂直, 单位体积内的粒子数为 $n$, 粒子与该平面碰撞后均不反弹，忽略空气阻力，不考虑粒子所受重力以及粒子间的相互作用。求粒子流对该平面所产生的压强 $p$ 。 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_997

For a satellite in a circular orbit around the Earth, which of the arrows in the figure below describe the direction of the velocity and the resultant force? [figure1] A: Velocity $=1$, Resultant force $=1$ B: Velocity $=1$, Resultant force $=2$ C: Velocity $=2$, Resultant force $=1$ D: Velocity $=3$, Resultant force $=4$

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: For a satellite in a circular orbit around the Earth, which of the arrows in the figure below describe the direction of the velocity and the resultant force? [figure1] A: Velocity $=1$, Resultant force $=1$ B: Velocity $=1$, Resultant force $=2$ C: Velocity $=2$, Resultant force $=1$ D: Velocity $=3$, Resultant force $=4$ 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].

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Astronomy

EN

multi-modal

Astronomy_740

宇航员驾驶宇宙飞船绕一星球做匀速圆周运动, 测得飞船线速度大小的二次方与轨道半径的倒数的关系图像如图中实线所示, 该图线 (直线) 的斜率为 $k$, 图中 $r_{0}$ (该星球的半径）为已知量。引力常量为 $G$, 下列说法正确的是 ( ) [图1] A: 该星球的密度为 $\frac{3 k}{4 \pi G r_{0}^{3}}$ B: 该星球自转的周期为 $\sqrt{\frac{r_{0}^{3}}{k}}$ C: 该星球表面的重力加速度大小为 $\frac{k}{r_{0}}$ D: 该星球的第一宇宙速度为 $\sqrt{\frac{2 k}{r_{0}}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 宇航员驾驶宇宙飞船绕一星球做匀速圆周运动, 测得飞船线速度大小的二次方与轨道半径的倒数的关系图像如图中实线所示, 该图线 (直线) 的斜率为 $k$, 图中 $r_{0}$ (该星球的半径）为已知量。引力常量为 $G$, 下列说法正确的是 ( ) [图1] A: 该星球的密度为 $\frac{3 k}{4 \pi G r_{0}^{3}}$ B: 该星球自转的周期为 $\sqrt{\frac{r_{0}^{3}}{k}}$ C: 该星球表面的重力加速度大小为 $\frac{k}{r_{0}}$ D: 该星球的第一宇宙速度为 $\sqrt{\frac{2 k}{r_{0}}}$ 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应为以下选项之一：[A, B, C, D]

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Astronomy

ZH

multi-modal

Astronomy_528

如图所示, $A$ 为地球表面赤道上的物体, $B$ 为轨道在赤道平面内的实验卫星, $C$ 为 在赤道上空的地球同步卫星, 已知卫星 $C$ 和卫星 $B$ 的轨道半径之比为 3: 1, 且两卫星的环绕方向相同, 下列说法正确的是 ( ) [图1] A: 卫星 $B 、 C$ 运行速度之比为 $3: 1$ B: 卫星 $B$ 的加速度大于物体 $A$ 的加速度 C: 同一物体在卫星 $B$ 中对支持物的压力比在卫星 $C$ 中大 D: 在卫星 $B$ 中一天内可看到 3 次日出

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 如图所示, $A$ 为地球表面赤道上的物体, $B$ 为轨道在赤道平面内的实验卫星, $C$ 为 在赤道上空的地球同步卫星, 已知卫星 $C$ 和卫星 $B$ 的轨道半径之比为 3: 1, 且两卫星的环绕方向相同, 下列说法正确的是 ( ) [图1] A: 卫星 $B 、 C$ 运行速度之比为 $3: 1$ B: 卫星 $B$ 的加速度大于物体 $A$ 的加速度 C: 同一物体在卫星 $B$ 中对支持物的压力比在卫星 $C$ 中大 D: 在卫星 $B$ 中一天内可看到 3 次日出 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

ZH

multi-modal

Astronomy_63

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, 则（ ） [图1] A: $\mathrm{M}$ 与 $\mathrm{N}$ 的密度相等 B: $Q$ 的质量是 $P$ 的 3 倍 C: $\mathrm{Q}$ 下落过程中的最大动能是 $\mathrm{P}$ 的 4 倍 D: Q 下落过程中弹簧的最大压缩量是 $\mathrm{P}$ 的 4 倍

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 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, 则（ ） [图1] A: $\mathrm{M}$ 与 $\mathrm{N}$ 的密度相等 B: $Q$ 的质量是 $P$ 的 3 倍 C: $\mathrm{Q}$ 下落过程中的最大动能是 $\mathrm{P}$ 的 4 倍 D: Q 下落过程中弹簧的最大压缩量是 $\mathrm{P}$ 的 4 倍 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_123

科学家在夏威夷利用红外望远镜设施发现了两颗近地小行星, 这两颗小行星富含金属, 且行星表面金属含量超过了 $85 \%$. 倘若有甲、乙两颗行星, 且各自卫星公转半径的三次方的倒数 $\frac{1}{r^{3}}$ 与公转角速度的平方 $\omega^{2}$ 的关系图像如图所示, 其中甲对应图线 $a$, 乙对应图线 $b$, 且甲、乙两行星的半径接近, 为方便分析认为两者相等, 下列说法正确的是 ( ) [图1] A: 甲行星的质量比乙行星大 B: 甲行星表面的卫星速度比乙行星小 C: 若甲、乙分别有一颗卫星 $A 、 B$, 且 $A 、 B$ 运行周期相同, 则 $A$ 的速度较大 D: 若甲、乙分别有一颗卫星 $C 、 D$, 且 $C 、 D$ 轨道半径相同, 则 $D$ 的向心加速度较大

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: 科学家在夏威夷利用红外望远镜设施发现了两颗近地小行星, 这两颗小行星富含金属, 且行星表面金属含量超过了 $85 \%$. 倘若有甲、乙两颗行星, 且各自卫星公转半径的三次方的倒数 $\frac{1}{r^{3}}$ 与公转角速度的平方 $\omega^{2}$ 的关系图像如图所示, 其中甲对应图线 $a$, 乙对应图线 $b$, 且甲、乙两行星的半径接近, 为方便分析认为两者相等, 下列说法正确的是 ( ) [图1] A: 甲行星的质量比乙行星大 B: 甲行星表面的卫星速度比乙行星小 C: 若甲、乙分别有一颗卫星 $A 、 B$, 且 $A 、 B$ 运行周期相同, 则 $A$ 的速度较大 D: 若甲、乙分别有一颗卫星 $C 、 D$, 且 $C 、 D$ 轨道半径相同, 则 $D$ 的向心加速度较大 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_480

据中新网报道, 中国自主研发的北斗卫星导航系统“北斗三号”第 17 颗卫星已于 2018 年 11 月 2 日在西昌卫星发射中心成功发射。该卫星是北斗三号全球导航系统的首颗地球同步轨道卫星, 也是北斗三号系统中功能最强、信号最多、承载最大、寿命最长的卫星。关于该卫星, 下列说法正确的是（） A: 它的发射速度一定小于 $11.2 \mathrm{~km} / \mathrm{s}$ B: 它运行的线速度一定不小于 $7.9 \mathrm{~km} / \mathrm{s}$ C: 它在由过渡轨道进入运行轨道时必须减速 D: 由于稀薄大气的影响, 如不加干预, 在运行一段时间后, 该卫星的速度可能会增加

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 据中新网报道, 中国自主研发的北斗卫星导航系统“北斗三号”第 17 颗卫星已于 2018 年 11 月 2 日在西昌卫星发射中心成功发射。该卫星是北斗三号全球导航系统的首颗地球同步轨道卫星, 也是北斗三号系统中功能最强、信号最多、承载最大、寿命最长的卫星。关于该卫星, 下列说法正确的是（） A: 它的发射速度一定小于 $11.2 \mathrm{~km} / \mathrm{s}$ B: 它运行的线速度一定不小于 $7.9 \mathrm{~km} / \mathrm{s}$ C: 它在由过渡轨道进入运行轨道时必须减速 D: 由于稀薄大气的影响, 如不加干预, 在运行一段时间后, 该卫星的速度可能会增加 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

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Astronomy

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text-only

Astronomy_622

如图所示, $a$ 为放在地球赤道上随地球表面一起转动的物体, $b$ 为处于地面附近近地轨道上的卫星, $c$ 是地球同步卫星, $d$ 是高空探测卫星, 若 $a 、 b 、 c 、 d$ 的质量相同,地球表面附近的重力加速度为 $g$. 则下列说法正确的是( ) [图1] A: $d$ 是三颗卫星中动能最小，机械能最大的 B: $c$ 距离地面的高度不是一确定值 C: $a$ 和 $b$ 的向心加速度都等于重力加速度 $g$ D: $a$ 的角速度最大

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, $a$ 为放在地球赤道上随地球表面一起转动的物体, $b$ 为处于地面附近近地轨道上的卫星, $c$ 是地球同步卫星, $d$ 是高空探测卫星, 若 $a 、 b 、 c 、 d$ 的质量相同,地球表面附近的重力加速度为 $g$. 则下列说法正确的是( ) [图1] A: $d$ 是三颗卫星中动能最小，机械能最大的 B: $c$ 距离地面的高度不是一确定值 C: $a$ 和 $b$ 的向心加速度都等于重力加速度 $g$ D: $a$ 的角速度最大 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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Astronomy_25

2017 年 8 月,我国调制望远镜一“慧眼"成功监测到了引力波源所在的天区。已知 A、 $\mathrm{B}$ 两个恒星靠着相互间的引力绕二者连线上的某点做匀速圆周运动. 在环绕过程中会辐射出引力波, 该引力波的频率与两星做圆周运动的频率具有相同的数量级. 通过观察测得 A 的质量为太阳质量的 29 倍, B 的质量为太阳质量的 36 倍,两星间的距离为 $2 \times$ $10^{5} \mathrm{~m}$. 取太阳的质量为 $2 \times 10^{30} \mathrm{~kg}, \mathrm{G}=6.7 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2}$, 则可估算出该引力波频率的数量级为 ( ) A: $10^{2} \mathrm{~Hz}$ B: $10^{4} \mathrm{~Hz}$ C: $10^{6} \mathrm{~Hz}$ D: $10^{8} \mathrm{~Hz}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 2017 年 8 月,我国调制望远镜一“慧眼"成功监测到了引力波源所在的天区。已知 A、 $\mathrm{B}$ 两个恒星靠着相互间的引力绕二者连线上的某点做匀速圆周运动. 在环绕过程中会辐射出引力波, 该引力波的频率与两星做圆周运动的频率具有相同的数量级. 通过观察测得 A 的质量为太阳质量的 29 倍, B 的质量为太阳质量的 36 倍,两星间的距离为 $2 \times$ $10^{5} \mathrm{~m}$. 取太阳的质量为 $2 \times 10^{30} \mathrm{~kg}, \mathrm{G}=6.7 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2}$, 则可估算出该引力波频率的数量级为 ( ) A: $10^{2} \mathrm{~Hz}$ B: $10^{4} \mathrm{~Hz}$ C: $10^{6} \mathrm{~Hz}$ D: $10^{8} \mathrm{~Hz}$ 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