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GAIR 62

Astronomy_745

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

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: Which one of these wavelengths is considered infrared radiation? A: 150 meters B: 150 millimeters C: 150 micrometers D: 150 nanometers You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

SC

Astronomy

EN

text-only

Astronomy_830

Assuming that the radius of the smaller star is 2 solar radii, what is the distance to the system? A: 75 parsecs B: 85 parsecs C: 100 parsecs D: 115 parsecs E: 150 parsecs

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: Assuming that the radius of the smaller star is 2 solar radii, what is the distance to the system? A: 75 parsecs B: 85 parsecs C: 100 parsecs D: 115 parsecs E: 150 parsecs 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_147

2020 年 11 月 24 日凌晨, 搭载嫦娥五号探测器的长征五号遥五运载火箭从文昌航天发射场顺利升空，12 月 17 日“嫦娥五号”返回器携带月球样品在预定区域安全着陆,在落地之前, 它在大气层打个“水漂”。现简化过程如图所示, 以地心为中心、半径为 $r_{1}$的圆周为大气层的边界, 忽略大气层外空气阻力。已知地球半径为 $R$, 返回器从 $o$ 点进入大气层，经 $a 、 b 、 c 、 d$ 回到地面，其中 $o 、 b 、 d$ 为轨道和大气层外边界的交点， $c$ 点到地心的距离为 $r_{2}$, 地球表面重力加速度为 $g$, 以下结论正确的是 ( ) [图1] A: 返回器在 $o$ 点动能大于在 $b$ 点的动能 B: 返回器在 $b$ 点动能大于在 $d$ 点的动能 C: 返回器在 $c$ 点的加速度大于 $\frac{g R^{2}}{r_{1}^{2}}$ D: 返回器通过 $a$ 点处于失重状态

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 11 月 24 日凌晨, 搭载嫦娥五号探测器的长征五号遥五运载火箭从文昌航天发射场顺利升空，12 月 17 日“嫦娥五号”返回器携带月球样品在预定区域安全着陆,在落地之前, 它在大气层打个“水漂”。现简化过程如图所示, 以地心为中心、半径为 $r_{1}$的圆周为大气层的边界, 忽略大气层外空气阻力。已知地球半径为 $R$, 返回器从 $o$ 点进入大气层，经 $a 、 b 、 c 、 d$ 回到地面，其中 $o 、 b 、 d$ 为轨道和大气层外边界的交点， $c$ 点到地心的距离为 $r_{2}$, 地球表面重力加速度为 $g$, 以下结论正确的是 ( ) [图1] A: 返回器在 $o$ 点动能大于在 $b$ 点的动能 B: 返回器在 $b$ 点动能大于在 $d$ 点的动能 C: 返回器在 $c$ 点的加速度大于 $\frac{g R^{2}}{r_{1}^{2}}$ D: 返回器通过 $a$ 点处于失重状态 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在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_13

人造地球卫星绕地球做匀速圆周运动, 例如卫星的周期增大到原来的 8 倍, 卫星仍做匀速圆周运动，则下列说法中正确的是（ ） A: 卫星的向心加速度增大到原来的 4 倍 B: 卫星的角速度减小到原来的 $1 / 4$ C: 卫星的动能将增大到原来的 4 倍 D: 卫星的线速度减小到原来的 $1 / 2$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 人造地球卫星绕地球做匀速圆周运动, 例如卫星的周期增大到原来的 8 倍, 卫星仍做匀速圆周运动，则下列说法中正确的是（ ） A: 卫星的向心加速度增大到原来的 4 倍 B: 卫星的角速度减小到原来的 $1 / 4$ C: 卫星的动能将增大到原来的 4 倍 D: 卫星的线速度减小到原来的 $1 / 2$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

text-only

Astronomy_616

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

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 已知地球半径为 $R$, 地球表面的重力加速度为 $g$ 。质量为 $m$ 的宇宙飞船在半径为 $2 R$的轨道 1 上绕地球中心 $O$ 做圆两运动。现飞船在轨道 1 的 $A$ 点加速到陏圆轨道 2 上,再在远地点 $B$ 点加速, 从而使飞船转移到半径为 $4 R$ 的轨道 3 上, 如图所示。若相距 $r$的两物体间引力势能为 $E_{\mathrm{p}}=-G \frac{M m}{r}$, 求: 飞船在轨道 2 上从 $A$ 点到 $B$ 点飞行的时间。 [图1] 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

EX

Astronomy

ZH

multi-modal

Astronomy_457

假设地球半径为 $R$, 地球表面的重力加速度为 $g_{0}$. “神舟九号”飞船沿距地球表面高度为 $3 R$ 的圆形轨道I运动, 到达轨道的 $A$ 点, 点火变轨进入粗圆轨道II, 到达轨道II的近地点 $B$ 再次点火进入近地轨道III绕地球做圆周运动. 下列判断正确的是 ( ) [图1] A: 飞船在轨道III跟轨道I的线速度大小之比为 $1: 2$ B: 飞船在轨道III跟轨道I的线速度大小之比为 $2: 1$ C: 飞船在轨道I绕地球运动一周所需的时间为 $2 \pi \sqrt{\frac{27 R}{g_{0}}}$ D: 飞船在轨道I绕地球运动一周所需的时间为 $16 \pi \sqrt{\frac{R}{g_{0}}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 假设地球半径为 $R$, 地球表面的重力加速度为 $g_{0}$. “神舟九号”飞船沿距地球表面高度为 $3 R$ 的圆形轨道I运动, 到达轨道的 $A$ 点, 点火变轨进入粗圆轨道II, 到达轨道II的近地点 $B$ 再次点火进入近地轨道III绕地球做圆周运动. 下列判断正确的是 ( ) [图1] A: 飞船在轨道III跟轨道I的线速度大小之比为 $1: 2$ B: 飞船在轨道III跟轨道I的线速度大小之比为 $2: 1$ C: 飞船在轨道I绕地球运动一周所需的时间为 $2 \pi \sqrt{\frac{27 R}{g_{0}}}$ D: 飞船在轨道I绕地球运动一周所需的时间为 $16 \pi \sqrt{\frac{R}{g_{0}}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_688

我国航天技术水平在世界处于领先地位，对于人造卫星的发射，有人提出了利用“地球隧道”发射人造卫星的构想：沿地球的一条弦挖一通道，在通道的两个出口处分别将等质量的待发射卫星部件同时释放，部件将在通道中间位置“碰撞组装”成卫星并静止下来; 另在通道的出口处由静止释放一个大质量物体，大质量物体会在通道与待发射的卫星碰撞, 只要物体质量相比卫星质量足够大, 卫星获得足够速度就会从对向通道口射出。 (以下计算中, 已知地球的质量为 $M_{0}$, 地球半径为 $R_{0}$, 引力常量为 $G$, 可忽略通道 $A B$的内径大小和地球自转影响。) 如图甲所示, 将一个质量为 $m_{0}$ 的质点置于质量分布均匀的球形天体内, 质点离球心 $O$ 的距离为 $r$ 。已知天体内部半径在 $r \sim R$ 之间的“球壳”部分（如甲示阴影部分）对质点的万有引力为零, 求质点所受万有引力的大小 $F_{\mathrm{r}}$ 。 [图1] 甲

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 我国航天技术水平在世界处于领先地位，对于人造卫星的发射，有人提出了利用“地球隧道”发射人造卫星的构想：沿地球的一条弦挖一通道，在通道的两个出口处分别将等质量的待发射卫星部件同时释放，部件将在通道中间位置“碰撞组装”成卫星并静止下来; 另在通道的出口处由静止释放一个大质量物体，大质量物体会在通道与待发射的卫星碰撞, 只要物体质量相比卫星质量足够大, 卫星获得足够速度就会从对向通道口射出。 (以下计算中, 已知地球的质量为 $M_{0}$, 地球半径为 $R_{0}$, 引力常量为 $G$, 可忽略通道 $A B$的内径大小和地球自转影响。) 如图甲所示, 将一个质量为 $m_{0}$ 的质点置于质量分布均匀的球形天体内, 质点离球心 $O$ 的距离为 $r$ 。已知天体内部半径在 $r \sim R$ 之间的“球壳”部分（如甲示阴影部分）对质点的万有引力为零, 求质点所受万有引力的大小 $F_{\mathrm{r}}$ 。 [图1] 甲 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_1201

On $21^{\text {st }}$ December 2020, Jupiter and Saturn formed a true spectacle in the southwestern sky just after sunset in the UK, separated by only $0.102^{\circ}$. This is close enough that when viewed through a telescope both planets and their moons could be seen in the same field of view (see Fig 4). When two planets occupy the same piece of sky it is known as a conjunction, and when it is Jupiter and Saturn it is known as a great conjunction (so named because they are the rarest of the naked-eye planet conjunctions). The reason it happens is because Jupiter's orbital velocity is higher than Saturn's and so as time goes on Jupiter catches up with and overtakes Saturn (at least as viewed from Earth), with the moment of overtaking corresponding to the conjunction. The process of the two getting closer and closer together has been seen in sky throughout the year (see Fig 5). [figure1] Figure 4: Left: The view of the planets the day before their closest approach, as captured by the 16 " telescope at the Institute of Astronomy, Cambridge. Credit: Robin Catchpole. Right: The view from Arizona on the day of the closest approach as viewed with the Lowell Discovery Telescope. Credit: Levine / Elbert / Bosh / Lowell Observatory. [figure2] Figure 5: Left: Demonstrating how Jupiter and Saturn have been getting closer and closer together over the last few months. Separations are given in degrees and arcminutes $\left(1 / 60^{\text {th }}\right.$ of a degree). Credit: Pete Lawrence. Right: The positions of the Earth, Jupiter and Saturn that were responsible for the 2020 great conjunction. The precise timing of the apparent alignment is clearly sensitive to where Earth is in its orbit. Credit: timeanddate.com. The time between conjunctions is known as the synodic period. Although this period will change slightly from conjunction to conjunction due to different lines of perspective as viewed from Earth (see Fig 5), we can work out the average time between great conjunctions by considering both planets travelling on circular coplanar orbits and ignoring the position of the Earth. For circular coplanar orbits the centre of Jupiter's disc would pass in front of the centre of Saturn's disc every conjunction, and hence have an angular separation of $\theta=0^{\circ}$ (it is measured from the centre of each disc). In practice, the planets follow elliptical orbits that are in planes inclined at different angles to each other. Fig 6 shows how this affects the real values for over 8000 years' worth of data, which along with different synodic periods between conjunctions makes it a difficult problem to solve precisely without a computer. However, after one synodic period Saturn has moved about $2 / 3$ of the way around its orbit, and so roughly every 3 synodic periods it is in a similar part of the sky. Consequently every third great conjunction follows a reasonably regular pattern which can be fit with a sinusoidal function. [figure3] Figure 6: Top: All great conjunctions from 1800 to 2300, calculated for the real celestial mechanics of the Solar System. We can see that each great conjunction belongs to one of three different series or tracks, with Track A indicated with orange circles, Track B with green squares, and Track C with blue triangles. Bottom: The same idea but extended over a much larger date range, up to $10000 \mathrm{AD}$. It is clear the distinct series form broadly sinusoidal patterns which can be used with the average synodic period to give rough predictions for the separations of great conjunctions. The opacity of points is related to each conjunction's angular separation from the Sun (where low opacity means close to the Sun, so it is harder for any observers to see). Credit: Nick Koukoufilippas, but inspired by the work of Steffen Thorsen and Graham Jones / Sky \& Telescope.a. In ideal observing conditions the two planets are far enough apart that they should be (just about) distinguishable to the naked eye, however to some observers in imperfect conditions they would appear as a single bright dot, brighter than either planet on its own. i. During the conjunction, the apparent magnitudes of Jupiter and Saturn were $m_{J}=-1.97$ and $m_{S}=0.63$, respectively (ignoring dimming by the atmosphere). What would be the apparent magnitude of the two planets if they appeared to an observer as a single point? [Hint: it is not simply $-1.97-0.63=-2.60$ ]

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: On $21^{\text {st }}$ December 2020, Jupiter and Saturn formed a true spectacle in the southwestern sky just after sunset in the UK, separated by only $0.102^{\circ}$. This is close enough that when viewed through a telescope both planets and their moons could be seen in the same field of view (see Fig 4). When two planets occupy the same piece of sky it is known as a conjunction, and when it is Jupiter and Saturn it is known as a great conjunction (so named because they are the rarest of the naked-eye planet conjunctions). The reason it happens is because Jupiter's orbital velocity is higher than Saturn's and so as time goes on Jupiter catches up with and overtakes Saturn (at least as viewed from Earth), with the moment of overtaking corresponding to the conjunction. The process of the two getting closer and closer together has been seen in sky throughout the year (see Fig 5). [figure1] Figure 4: Left: The view of the planets the day before their closest approach, as captured by the 16 " telescope at the Institute of Astronomy, Cambridge. Credit: Robin Catchpole. Right: The view from Arizona on the day of the closest approach as viewed with the Lowell Discovery Telescope. Credit: Levine / Elbert / Bosh / Lowell Observatory. [figure2] Figure 5: Left: Demonstrating how Jupiter and Saturn have been getting closer and closer together over the last few months. Separations are given in degrees and arcminutes $\left(1 / 60^{\text {th }}\right.$ of a degree). Credit: Pete Lawrence. Right: The positions of the Earth, Jupiter and Saturn that were responsible for the 2020 great conjunction. The precise timing of the apparent alignment is clearly sensitive to where Earth is in its orbit. Credit: timeanddate.com. The time between conjunctions is known as the synodic period. Although this period will change slightly from conjunction to conjunction due to different lines of perspective as viewed from Earth (see Fig 5), we can work out the average time between great conjunctions by considering both planets travelling on circular coplanar orbits and ignoring the position of the Earth. For circular coplanar orbits the centre of Jupiter's disc would pass in front of the centre of Saturn's disc every conjunction, and hence have an angular separation of $\theta=0^{\circ}$ (it is measured from the centre of each disc). In practice, the planets follow elliptical orbits that are in planes inclined at different angles to each other. Fig 6 shows how this affects the real values for over 8000 years' worth of data, which along with different synodic periods between conjunctions makes it a difficult problem to solve precisely without a computer. However, after one synodic period Saturn has moved about $2 / 3$ of the way around its orbit, and so roughly every 3 synodic periods it is in a similar part of the sky. Consequently every third great conjunction follows a reasonably regular pattern which can be fit with a sinusoidal function. [figure3] Figure 6: Top: All great conjunctions from 1800 to 2300, calculated for the real celestial mechanics of the Solar System. We can see that each great conjunction belongs to one of three different series or tracks, with Track A indicated with orange circles, Track B with green squares, and Track C with blue triangles. Bottom: The same idea but extended over a much larger date range, up to $10000 \mathrm{AD}$. It is clear the distinct series form broadly sinusoidal patterns which can be used with the average synodic period to give rough predictions for the separations of great conjunctions. The opacity of points is related to each conjunction's angular separation from the Sun (where low opacity means close to the Sun, so it is harder for any observers to see). Credit: Nick Koukoufilippas, but inspired by the work of Steffen Thorsen and Graham Jones / Sky \& Telescope. problem: a. In ideal observing conditions the two planets are far enough apart that they should be (just about) distinguishable to the naked eye, however to some observers in imperfect conditions they would appear as a single bright dot, brighter than either planet on its own. i. During the conjunction, the apparent magnitudes of Jupiter and Saturn were $m_{J}=-1.97$ and $m_{S}=0.63$, respectively (ignoring dimming by the atmosphere). What would be the apparent magnitude of the two planets if they appeared to an observer as a single point? [Hint: it is not simply $-1.97-0.63=-2.60$ ] 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_699

2021 年 9 月 17 日, “神舟十二号”返回舱在东风着陆场安全降落。返回舱从工作轨道I返回地面的运动轨迹如图, 椭圆轨道II与圆轨道I、III分别相切于 $P 、 Q$ 两点, 返回舱从轨道III上适当位置减速后进入大气层, 最后在东风着陆场着陆。下列说法正确的是 [图1] A: 返回舱在I轨道上 $P$ 需要向运动方向的反方向喷气进入II轨道 B: 返回舱在II轨道上运动的周期小于返回舱在III轨道上运动的周期 C: 返回舱在III轨道上 $Q$ 点的速度的大小大于II轨道上 $P$ 点速度的大小 D: 返回舱在I轨道上经过 $P$ 点时的加速度等于在II轨道上经过 $P$ 点时的加速度

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 2021 年 9 月 17 日, “神舟十二号”返回舱在东风着陆场安全降落。返回舱从工作轨道I返回地面的运动轨迹如图, 椭圆轨道II与圆轨道I、III分别相切于 $P 、 Q$ 两点, 返回舱从轨道III上适当位置减速后进入大气层, 最后在东风着陆场着陆。下列说法正确的是 [图1] A: 返回舱在I轨道上 $P$ 需要向运动方向的反方向喷气进入II轨道 B: 返回舱在II轨道上运动的周期小于返回舱在III轨道上运动的周期 C: 返回舱在III轨道上 $Q$ 点的速度的大小大于II轨道上 $P$ 点速度的大小 D: 返回舱在I轨道上经过 $P$ 点时的加速度等于在II轨道上经过 $P$ 点时的加速度 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_169

利用水流和太阳能发电, 可以为人类提供清洁能源。已知太阳光垂直照射到地面上时的辐射功率 $P_{0}=1.0 \times 10^{3} \mathrm{~W} / \mathrm{m}^{2}$, 地球表面的重力加速度取 $g=10 \mathrm{~m} / \mathrm{s}^{2}$, 水的密度 $\rho=1.0 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ 。 若利用太阳能发电, 需要发射一颗卫星到地球同步轨道上, 然后通过微波持续不断地将电能输送到地面, 这样就建成了宇宙太阳能发电站。已知地球同步轨道半径约为地球半径的 $2 \sqrt{11}$ 倍。求卫星在地球同步轨道上向心加速度的大小;

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个数值。 问题: 利用水流和太阳能发电, 可以为人类提供清洁能源。已知太阳光垂直照射到地面上时的辐射功率 $P_{0}=1.0 \times 10^{3} \mathrm{~W} / \mathrm{m}^{2}$, 地球表面的重力加速度取 $g=10 \mathrm{~m} / \mathrm{s}^{2}$, 水的密度 $\rho=1.0 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ 。 若利用太阳能发电, 需要发射一颗卫星到地球同步轨道上, 然后通过微波持续不断地将电能输送到地面, 这样就建成了宇宙太阳能发电站。已知地球同步轨道半径约为地球半径的 $2 \sqrt{11}$ 倍。求卫星在地球同步轨道上向心加速度的大小; 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 请记住，你的答案应以$\mathrm{~m} / \mathrm{s}^{2}$为单位计算，但在给出最终答案时，请不要包含单位。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。

NV

Astronomy

ZH

text-only

Astronomy_153

理论上已经证明, 质量分布均匀的球壳对壳内物体的万有引力为零。现假设地球是一半径为 $R$ 、质量分布均匀的实心球体, 将一个铁球分别放在地面以下 $\frac{R}{3}$ 深处和放在地面上方 $\frac{R}{3}$ 高度处, 则物体在两处的重力加速度之比为（） A: $32: 27$ B: $9: 8$ C: $81: 64$ D: $4: 3$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 理论上已经证明, 质量分布均匀的球壳对壳内物体的万有引力为零。现假设地球是一半径为 $R$ 、质量分布均匀的实心球体, 将一个铁球分别放在地面以下 $\frac{R}{3}$ 深处和放在地面上方 $\frac{R}{3}$ 高度处, 则物体在两处的重力加速度之比为（） A: $32: 27$ B: $9: 8$ C: $81: 64$ D: $4: 3$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

text-only

Astronomy_1218

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: ii. The new radius 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: ii. The new radius 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 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_991

Currently, Polaris is very close to the north celestial pole (the projection of the Earth's rotational axis on the sky) and so all other stars appear to rotate around it. However, this axis is drawing out a large circle in the sky with an angular radius of $23.44^{\circ}$ so Polaris will only temporarily be the pole star (see Figure 2). This precession of the rotational axis is mainly driven by the gravitational pull of the Moon and the Sun. [figure1] Figure 2: Top left: The Earth's rotational axis itself rotates slowly (white circle), in what is known as axial precession. Credit: David Battisti / University of Washington. Top right: Due to precession, the pole star has changed over time. About 5000 years ago, the star Thuban in the constellation of Draco was the pole star. Credit: Richard W. Pogge / Ohio State University. Bottom: The position of the Sun at the spring equinox (where the celestial equator meets the ecliptic) has also changed over the same period, moving from Aries to Pisces. Credit: Guy Ottewell / Universal Workshop. Another consequence is that the position of the Sun at the equinoxes varies slightly, moving slowly westwards. This gives rise to two definitions of a year: - a sidereal year (the time taken for the Earth to orbit the Sun once with respect to the background stars) $=365.256363$ days - a tropical year (the time taken for the Sun to return to the same position in the cycle of the seasons) $=365.242190$ days The Gregorian calendar is a 400-year cycle with a system of leap years. The rule is: "every year that is exactly divisible by four is a leap year, except for years that are exactly divisible by 100 , but these centurial years are leap years if they are exactly divisible by 400. ." Assuming the rate of axial precession remains constant, work out the time (in sidereal years) to complete a whole precessional cycle.

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: Currently, Polaris is very close to the north celestial pole (the projection of the Earth's rotational axis on the sky) and so all other stars appear to rotate around it. However, this axis is drawing out a large circle in the sky with an angular radius of $23.44^{\circ}$ so Polaris will only temporarily be the pole star (see Figure 2). This precession of the rotational axis is mainly driven by the gravitational pull of the Moon and the Sun. [figure1] Figure 2: Top left: The Earth's rotational axis itself rotates slowly (white circle), in what is known as axial precession. Credit: David Battisti / University of Washington. Top right: Due to precession, the pole star has changed over time. About 5000 years ago, the star Thuban in the constellation of Draco was the pole star. Credit: Richard W. Pogge / Ohio State University. Bottom: The position of the Sun at the spring equinox (where the celestial equator meets the ecliptic) has also changed over the same period, moving from Aries to Pisces. Credit: Guy Ottewell / Universal Workshop. Another consequence is that the position of the Sun at the equinoxes varies slightly, moving slowly westwards. This gives rise to two definitions of a year: - a sidereal year (the time taken for the Earth to orbit the Sun once with respect to the background stars) $=365.256363$ days - a tropical year (the time taken for the Sun to return to the same position in the cycle of the seasons) $=365.242190$ days The Gregorian calendar is a 400-year cycle with a system of leap years. The rule is: "every year that is exactly divisible by four is a leap year, except for years that are exactly divisible by 100 , but these centurial years are leap years if they are exactly divisible by 400. ." Assuming the rate of axial precession remains constant, work out the time (in sidereal years) to complete a whole precessional cycle. 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_1095

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.c. Show that for a disc of uniform entropy, $\Gamma_{=} \Gamma_{L}+\Gamma_{\mathrm{C}}=(5.76-(5.11+2.33 \gamma) \alpha) \Gamma_{0}$.

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: c. Show that for a disc of uniform entropy, $\Gamma_{=} \Gamma_{L}+\Gamma_{\mathrm{C}}=(5.76-(5.11+2.33 \gamma) \alpha) \Gamma_{0}$. 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_890

An astronomer takes a spectrum of a galaxy and observes that the hydrogen-alpha emission line is at a wavelength of 721.9 nanometers. In a laboratory on Earth, this same emission line is observed at a wavelength of 656.3 nanometers. Approximately what is the (proper) distance to this galaxy? A: $66 \mathrm{Mpc}$ B: $430 \mathrm{Mpc}$ C: $480 \mathrm{Mpc}$ D: $3900 \mathrm{Mpc}$ E: $4700 \mathrm{Mpc}$

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: An astronomer takes a spectrum of a galaxy and observes that the hydrogen-alpha emission line is at a wavelength of 721.9 nanometers. In a laboratory on Earth, this same emission line is observed at a wavelength of 656.3 nanometers. Approximately what is the (proper) distance to this galaxy? A: $66 \mathrm{Mpc}$ B: $430 \mathrm{Mpc}$ C: $480 \mathrm{Mpc}$ D: $3900 \mathrm{Mpc}$ E: $4700 \mathrm{Mpc}$ You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D, E].

SC

Astronomy

EN

text-only

Astronomy_317

已知地球的质量是月球质量的 81 倍, 地球半径大约是月球半径的 4 倍, 不考虑地球、月球自转的影响, 以上数据可推算出 [ ] A: 地球表面的重力加速度与月球表面重力加速度之比为 9: 16 B: 地球的平均密度与月球的平均密度之比为 $9: 8$ C: 靠近地球表面沿圆轨道运动的航天器的周期与靠近月球表面沿圆轨道运行的航天器的周期之比约为 $8: 9$ D: 靠近地球表面沿圆轨道运行的航天器的线速度与靠近月球表面沿圆轨道运行的航天器的线速度之比约为 $81: 4$

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: 已知地球的质量是月球质量的 81 倍, 地球半径大约是月球半径的 4 倍, 不考虑地球、月球自转的影响, 以上数据可推算出 [ ] A: 地球表面的重力加速度与月球表面重力加速度之比为 9: 16 B: 地球的平均密度与月球的平均密度之比为 $9: 8$ C: 靠近地球表面沿圆轨道运动的航天器的周期与靠近月球表面沿圆轨道运行的航天器的周期之比约为 $8: 9$ D: 靠近地球表面沿圆轨道运行的航天器的线速度与靠近月球表面沿圆轨道运行的航天器的线速度之比约为 $81: 4$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

text-only

Astronomy_837

Assume that the smaller star in the above binary star system is brighter than the larger star. What is the ratio of the radius of the smaller star to the radius of the larger star? A: 0.21 B: 0.76 C: 0.82 D: 0.95 E: 0.98

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: Assume that the smaller star in the above binary star system is brighter than the larger star. What is the ratio of the radius of the smaller star to the radius of the larger star? A: 0.21 B: 0.76 C: 0.82 D: 0.95 E: 0.98 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_121

如图所示, 有 $\mathrm{A} 、 \mathrm{~B}$ 两颗行星绕同一恒星 $\mathrm{O}$ 做圆周运动, 运行方向相同。 $\mathrm{A}$ 行星的周期为 $T_{1}, \mathrm{~B}$ 行星的周期为 $T_{2}$, 在某一时刻两行星相距最近, 则 ( ) [图1] A: 经过时间 $t=\frac{T_{1} T_{2}}{T_{2}-T_{1}}$, 两行星将再次相距最近 B: 经过时间 $t=T_{1}+T_{2}$, 两行星将再次相距最近 C: 经过时间 $t=\frac{n T_{1} T_{2}}{2\left(T_{2}-T_{1}\right)}(n=1,3,5, \ldots)$, 两行星相距最远 D: 经过时间 $t=\frac{n\left(T_{1}+T_{2}\right)}{2}(n=1,3,5, \ldots)$, 两行星相距最远

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 如图所示, 有 $\mathrm{A} 、 \mathrm{~B}$ 两颗行星绕同一恒星 $\mathrm{O}$ 做圆周运动, 运行方向相同。 $\mathrm{A}$ 行星的周期为 $T_{1}, \mathrm{~B}$ 行星的周期为 $T_{2}$, 在某一时刻两行星相距最近, 则 ( ) [图1] A: 经过时间 $t=\frac{T_{1} T_{2}}{T_{2}-T_{1}}$, 两行星将再次相距最近 B: 经过时间 $t=T_{1}+T_{2}$, 两行星将再次相距最近 C: 经过时间 $t=\frac{n T_{1} T_{2}}{2\left(T_{2}-T_{1}\right)}(n=1,3,5, \ldots)$, 两行星相距最远 D: 经过时间 $t=\frac{n\left(T_{1}+T_{2}\right)}{2}(n=1,3,5, \ldots)$, 两行星相距最远 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_709

2018 年 12 月 27 日, 我国北斗卫星导航系统开始提供全球服务, 标志着北斗系统正式迈入全球时代。覆盖全球的北斗卫星导航系统是由静止轨道卫星 (即地球同步卫星)和非静止轨道卫星共 35 颗组成的。卫星绕地球近似做匀速圆周运动。已知其中一颗地球同步卫星距离地球表面的高度为 $h$, 地球质量为 $M$, 地球半径为 $R$, 引力常量为 $G$ 。 如图所示, $O$ 点为地球的球心, $P$ 点处有一颗地球同步卫星, $P$ 点所在的虚线圆轨道为同步卫星绕地球运动的轨道。已知 $h=5.6 R$ 。忽略大气等一切影响因素, 请论证说明要使卫星通讯覆盖全球, 至少需要几颗地球同步卫星。 $\left(\cos 81^{\circ} \approx 0.15\right.$, $\sin 81^{\circ} \approx 0.99$ ) [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: 2018 年 12 月 27 日, 我国北斗卫星导航系统开始提供全球服务, 标志着北斗系统正式迈入全球时代。覆盖全球的北斗卫星导航系统是由静止轨道卫星 (即地球同步卫星)和非静止轨道卫星共 35 颗组成的。卫星绕地球近似做匀速圆周运动。已知其中一颗地球同步卫星距离地球表面的高度为 $h$, 地球质量为 $M$, 地球半径为 $R$, 引力常量为 $G$ 。 如图所示, $O$ 点为地球的球心, $P$ 点处有一颗地球同步卫星, $P$ 点所在的虚线圆轨道为同步卫星绕地球运动的轨道。已知 $h=5.6 R$ 。忽略大气等一切影响因素, 请论证说明要使卫星通讯覆盖全球, 至少需要几颗地球同步卫星。 $\left(\cos 81^{\circ} \approx 0.15\right.$, $\sin 81^{\circ} \approx 0.99$ ) [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，你的答案应以颗为单位计算，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_366

如图所示, 地球卫星开始在圆形低轨道 1 运行, 在 $P$ 处点火后, 地球卫星沿椭圆轨道 2 运行, 在远地点 $Q$ 再次点火, 将地球卫星送入更高的圆形轨道 3. 若地球卫星在 $1 、 3$ 轨道上运行的速率分别为 $v_{1} 、 v_{3}$, 在 2 轨道上经过 $P 、 Q$ 处的速率分别为 $v_{2 P} 、 v_{2 Q}$,则 [图1] A: $v_{3}<v_{1}$ B: $v_{2 P}<v_{2 Q}$ C: $v_{3}>v_{2 O}$ D: $v_{1}>v_{2 P}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 如图所示, 地球卫星开始在圆形低轨道 1 运行, 在 $P$ 处点火后, 地球卫星沿椭圆轨道 2 运行, 在远地点 $Q$ 再次点火, 将地球卫星送入更高的圆形轨道 3. 若地球卫星在 $1 、 3$ 轨道上运行的速率分别为 $v_{1} 、 v_{3}$, 在 2 轨道上经过 $P 、 Q$ 处的速率分别为 $v_{2 P} 、 v_{2 Q}$,则 [图1] A: $v_{3}<v_{1}$ B: $v_{2 P}<v_{2 Q}$ C: $v_{3}>v_{2 O}$ D: $v_{1}>v_{2 P}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在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_196

火星被认为是太阳系中最有可能存在地外生命的行星, 对人类来说充满着神奇! 2021 年 2 月, 包括我国“天问一号”在内的国际三个火星探测器全部抵达火星。若火星的密度为 $\rho$, 火星的一颗天然卫星绕火星做匀速圆周运动, 其线速度为 $v$, 运行周期为 $T$ 。已知引力常量为 $G$, 则可以求得 ( ) A: 火星的半径 B: 火星的质量 C: 天然卫星的质量 D: 天然卫星距离火星表面的高度

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 火星被认为是太阳系中最有可能存在地外生命的行星, 对人类来说充满着神奇! 2021 年 2 月, 包括我国“天问一号”在内的国际三个火星探测器全部抵达火星。若火星的密度为 $\rho$, 火星的一颗天然卫星绕火星做匀速圆周运动, 其线速度为 $v$, 运行周期为 $T$ 。已知引力常量为 $G$, 则可以求得 ( ) A: 火星的半径 B: 火星的质量 C: 天然卫星的质量 D: 天然卫星距离火星表面的高度 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

text-only

Astronomy_1166

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): i. Find the distance from the centre of the Earth to point $C$, and hence the semi-major axes of both ellipses.

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

NV

Astronomy

EN

multi-modal

Astronomy_28

有 $a 、 b 、 c 、 d$ 四颗地球卫星: $a$ 还未发射, 在地球赤道上随地球表面一起转动, $b$在地球的近地圆轨道上正常运行, $c$ 是地球同步卫星, $d$ 是高空探测卫星, 各卫星排列位置如图, 则下列说法正确的是（） [图1] A: 向心加速度大小关系是: $a_{b}>a_{c}>a_{d}>a_{a}$, 速度大小关系是: $v_{a}>v_{b}>v_{c}>v_{d}$ B: 在相同时间内 $b$ 转过的弧长最长, $a 、 c$ 转过的弧长对应的角度相等 C: $c$ 在 4 小时内转过的圆心角是 $\frac{\pi}{2}, a$ 在 2 小时内转过的圆心角是 $\frac{\pi}{6}$ D: $b$ 的周期一定小于 $d$ 的周期， $d$ 的周期一定大于 24 小时

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: 有 $a 、 b 、 c 、 d$ 四颗地球卫星: $a$ 还未发射, 在地球赤道上随地球表面一起转动, $b$在地球的近地圆轨道上正常运行, $c$ 是地球同步卫星, $d$ 是高空探测卫星, 各卫星排列位置如图, 则下列说法正确的是（） [图1] A: 向心加速度大小关系是: $a_{b}>a_{c}>a_{d}>a_{a}$, 速度大小关系是: $v_{a}>v_{b}>v_{c}>v_{d}$ B: 在相同时间内 $b$ 转过的弧长最长, $a 、 c$ 转过的弧长对应的角度相等 C: $c$ 在 4 小时内转过的圆心角是 $\frac{\pi}{2}, a$ 在 2 小时内转过的圆心角是 $\frac{\pi}{6}$ D: $b$ 的周期一定小于 $d$ 的周期， $d$ 的周期一定大于 24 小时 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在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_1190

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.calculate the mean difference in the distance between the apogee and perigee (The data given in this question allows the mean orbital parameters to be calculated. Note that in reality, perturbations in the lunar orbit mean that the perigee and apogee continually change over the course of the year).

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. 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: calculate the mean difference in the distance between the apogee and perigee (The data given in this question allows the mean orbital parameters to be calculated. Note that in reality, perturbations in the lunar orbit mean that the perigee and apogee continually change over the course of the year). All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of km, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

text-only

Astronomy_1175

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 $$ v^{2}=G M\left(\frac{2}{r}-\frac{1}{a}\right) $$a. Given the Sun's composition has hydrogen fraction, $X=0.72$, helium fraction $Y=0.26$ and 'metals' (i.e. any element lithium and heavier) fraction $Z=0.02$, estimate the temperature at the centre 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: 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: a. Given the Sun's composition has hydrogen fraction, $X=0.72$, helium fraction $Y=0.26$ and 'metals' (i.e. any element lithium and heavier) fraction $Z=0.02$, estimate the temperature at the centre 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 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.

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Astronomy

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multi-modal

Astronomy_411

2013 年 12 月 2 日 1 时 30 分, “嫦娥三号”探测器由长征三号乙运载火箭从西昌卫星发射中心发射, 首次实现月球软着陆和月面巡视勘察. 嫦娥三号的飞行轨道示意图如图所示. 假设“嫦娥三号”在环月段圆轨道和椭圆轨道上运动时，只受到月球的万有引力. 则 ( ) [图1] A: 若已知嫦娥三号环月段圆轨道的半径、运动周期和引力常量, 则可以计算出月球的密度 B: 嫦娥三号由环月段圆轨道变轨进入环月段椭圆轨道时, 应让发动机点火使其加速 C: 嫦娥三号在环月段椭圆轨道上 $\mathrm{P}$ 点的动能大于 $\mathrm{Q}$ 点的动能 D: 嫦娥三号在动力下降阶段, 其引力势能减小

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: 2013 年 12 月 2 日 1 时 30 分, “嫦娥三号”探测器由长征三号乙运载火箭从西昌卫星发射中心发射, 首次实现月球软着陆和月面巡视勘察. 嫦娥三号的飞行轨道示意图如图所示. 假设“嫦娥三号”在环月段圆轨道和椭圆轨道上运动时，只受到月球的万有引力. 则 ( ) [图1] A: 若已知嫦娥三号环月段圆轨道的半径、运动周期和引力常量, 则可以计算出月球的密度 B: 嫦娥三号由环月段圆轨道变轨进入环月段椭圆轨道时, 应让发动机点火使其加速 C: 嫦娥三号在环月段椭圆轨道上 $\mathrm{P}$ 点的动能大于 $\mathrm{Q}$ 点的动能 D: 嫦娥三号在动力下降阶段, 其引力势能减小 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

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multi-modal

Astronomy_1032

Figure 4 below is a composite image which depicts a transit of the International Space Station (ISS) across the disc of the Sun. The image comprises 26 individual photographs which were taken at regular time intervals during the transit. The total duration of the transit was less than one second. In this question we will ignore any effects caused by the rotation of the Earth. [figure1] Figure 4: A composite of a selection of the frames taken with a high-speed camera of a transit of the ISS in front of the Sun, taken from Northamptonshire at 10:22 BST on $17^{\text {th }}$ June 2022. Credit: Jamie Cooper Photography The orbital period of the ISS is approximately $93 \mathrm{~min}$. Estimate the frame rate of the camera used to photograph the transit.

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: Figure 4 below is a composite image which depicts a transit of the International Space Station (ISS) across the disc of the Sun. The image comprises 26 individual photographs which were taken at regular time intervals during the transit. The total duration of the transit was less than one second. In this question we will ignore any effects caused by the rotation of the Earth. [figure1] Figure 4: A composite of a selection of the frames taken with a high-speed camera of a transit of the ISS in front of the Sun, taken from Northamptonshire at 10:22 BST on $17^{\text {th }}$ June 2022. Credit: Jamie Cooper Photography The orbital period of the ISS is approximately $93 \mathrm{~min}$. Estimate the frame rate of the camera used to photograph the transit. 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 fps, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

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Astronomy

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multi-modal

Astronomy_733

如图所示, 有两颗卫星绕某星球做椭圆轨道运动, 两颗卫星的近地点均与星球表面很近 (可视为相切), 卫星 1 和卫星 2 的轨道远地点到星球表面的最近距离分别为 $h_{1} 、 h_{2}$, 卫星 1 和卫星 2 的环绕周期之比为 $k$ 。忽略星球自转的影响, 已知引力常量为 $G$, 星球表面的重力加速度为 $g_{c}$ 。则星球的平均密度为 ( ) [图1] A: $\frac{3 g_{c}\left(1-k^{\frac{2}{3}}\right)}{2 \pi G\left(h_{2} k^{\frac{2}{3}}-h_{1}\right)}$ B: $\frac{3 g_{c}\left(1-k^{\frac{3}{2}}\right)}{2 \pi G\left(h_{2} k^{\frac{3}{2}}-h_{1}\right)}$ C: $\frac{3 g_{c}\left(1-k^{\frac{3}{2}}\right)}{4 \pi G\left(h_{2} k^{\frac{3}{2}}-h_{1}\right)}$ D: $\frac{3 g_{c}\left(1-k^{\frac{2}{3}}\right)}{4 \pi G\left(h_{2} k^{\frac{2}{3}}-h_{1}\right)}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 如图所示, 有两颗卫星绕某星球做椭圆轨道运动, 两颗卫星的近地点均与星球表面很近 (可视为相切), 卫星 1 和卫星 2 的轨道远地点到星球表面的最近距离分别为 $h_{1} 、 h_{2}$, 卫星 1 和卫星 2 的环绕周期之比为 $k$ 。忽略星球自转的影响, 已知引力常量为 $G$, 星球表面的重力加速度为 $g_{c}$ 。则星球的平均密度为 ( ) [图1] A: $\frac{3 g_{c}\left(1-k^{\frac{2}{3}}\right)}{2 \pi G\left(h_{2} k^{\frac{2}{3}}-h_{1}\right)}$ B: $\frac{3 g_{c}\left(1-k^{\frac{3}{2}}\right)}{2 \pi G\left(h_{2} k^{\frac{3}{2}}-h_{1}\right)}$ C: $\frac{3 g_{c}\left(1-k^{\frac{3}{2}}\right)}{4 \pi G\left(h_{2} k^{\frac{3}{2}}-h_{1}\right)}$ D: $\frac{3 g_{c}\left(1-k^{\frac{2}{3}}\right)}{4 \pi G\left(h_{2} k^{\frac{2}{3}}-h_{1}\right)}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy_889

Estimate the mass of a globular cluster with a radial velocity dispersion $\sigma_{r}=16.2 \mathrm{~km} / \mathrm{s}$. The cluster has an angular diameter of $\theta=3.56^{\prime}$ and is a distance $d=9630$ pc away from us. A: $6.05 \times 10^{35} \mathrm{~kg}$ B: $9.71 \times 10^{35} \mathrm{~kg}$ C: $1.01 \times 10^{36} \mathrm{~kg}$ D: $3.03 \times 10^{36} \mathrm{~kg}$ E: $5.96 \times 10^{36} \mathrm{~kg}$

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: Estimate the mass of a globular cluster with a radial velocity dispersion $\sigma_{r}=16.2 \mathrm{~km} / \mathrm{s}$. The cluster has an angular diameter of $\theta=3.56^{\prime}$ and is a distance $d=9630$ pc away from us. A: $6.05 \times 10^{35} \mathrm{~kg}$ B: $9.71 \times 10^{35} \mathrm{~kg}$ C: $1.01 \times 10^{36} \mathrm{~kg}$ D: $3.03 \times 10^{36} \mathrm{~kg}$ E: $5.96 \times 10^{36} \mathrm{~kg}$ 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_200

如图所示, 有一质量为 $M$, 半径为 $R$, 密度均匀的球体, 在距离球心 $O$ 为 $2 R$ 的地方有一质量为 $m$ 的质点, 现从 $M$ 中挖去一半径为 $\frac{R}{2}$ 的球体, 试求: 若在挖空部分填满另外一种密度为原来 2 倍的物质, 求填充后的实心体对质点 $m$的引力大小。 [图1]

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 如图所示, 有一质量为 $M$, 半径为 $R$, 密度均匀的球体, 在距离球心 $O$ 为 $2 R$ 的地方有一质量为 $m$ 的质点, 现从 $M$ 中挖去一半径为 $\frac{R}{2}$ 的球体, 试求: 若在挖空部分填满另外一种密度为原来 2 倍的物质, 求填充后的实心体对质点 $m$的引力大小。 [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

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multi-modal

Astronomy_912

For which of these lines of latitude will a vertical stick in the ground have no shadow at local midday on $21^{\text {st }}$ December 2021 ? A: Tropic of Cancer B: Equator C: Tropic of Capricorn D: Antarctic Circle

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 which of these lines of latitude will a vertical stick in the ground have no shadow at local midday on $21^{\text {st }}$ December 2021 ? A: Tropic of Cancer B: Equator C: Tropic of Capricorn D: Antarctic Circle 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_994

On $5^{\text {th }}$ September 2016, the Rosetta mission has finally found the Philae lander on Comet 67P/Churyumov-Gerasimenko. Considering that Philae ( $1 \times 1 \times 1 \mathrm{~m})$ appeared in an image from the high-resolution camera (with $2048 \times 2048$ pixels and field of view $2.2^{\circ} \times 2.2^{\circ}$ ) as $25 \times 25$ pixels, from what distance did Rosetta manage to image Philae? A: $1.1 \mathrm{~km}$ B: $2.1 \mathrm{~km}$ C: $12.2 \mathrm{~km}$ D: $26.8 \mathrm{~km}$

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: On $5^{\text {th }}$ September 2016, the Rosetta mission has finally found the Philae lander on Comet 67P/Churyumov-Gerasimenko. Considering that Philae ( $1 \times 1 \times 1 \mathrm{~m})$ appeared in an image from the high-resolution camera (with $2048 \times 2048$ pixels and field of view $2.2^{\circ} \times 2.2^{\circ}$ ) as $25 \times 25$ pixels, from what distance did Rosetta manage to image Philae? A: $1.1 \mathrm{~km}$ B: $2.1 \mathrm{~km}$ C: $12.2 \mathrm{~km}$ D: $26.8 \mathrm{~km}$ You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

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Astronomy

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text-only

Astronomy_190

2019 年 1 月 3 日, 我国“嫦娥四号”探测器在月球背面成功着陆并发回大量月背影 像. 如图所示为位于月球背面的“嫦娥四号”探测器 $A$ 通过“鹊桥”中继站 $B$ 向地球传输电磁波信息的示意图. 拉格朗日 $L_{2}$ 点位于地月连线延长线上, “鹊桥”的运动可看成如下两种运动的合运动: 一是在地球和月球引力共同作用下, “鹊桥”在 $L_{2}$ 点附近与月球以相同的周期 $T_{0}$ 一起绕地球做匀速圆周运动; 二是在与地月连线垂直的平面内绕 $L_{2}$ 点做匀速圆周运动. 已知地球的质量为月球质量的 $n$ 倍, 地球到 $L_{2}$ 点的距离为月球到 $L_{2}$ 点的距离的 $k$ 倍, 地球半径、月球半径以及“鹊桥”绕 $L_{2}$ 点做匀速圆周运动的半径均远小于月球到 $L_{2}$ 点的距离 (提示: “鹊桥”绕 $L_{2}$ 点做匀速圆周运动的向心力由地球和月球对其引力在过 $L_{2}$ 点与地月连线垂直的平面内的分量提供).[图1] 试推导“鹊桥”绕 $L_{2}$ 点做匀速圆周运动的周期 $T$ 的（近似）表达式. 若 $k=7, n=81$, $T_{0}=27.3$ 天, 求出 $T$ 的天数（取整数）.

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: 2019 年 1 月 3 日, 我国“嫦娥四号”探测器在月球背面成功着陆并发回大量月背影 像. 如图所示为位于月球背面的“嫦娥四号”探测器 $A$ 通过“鹊桥”中继站 $B$ 向地球传输电磁波信息的示意图. 拉格朗日 $L_{2}$ 点位于地月连线延长线上, “鹊桥”的运动可看成如下两种运动的合运动: 一是在地球和月球引力共同作用下, “鹊桥”在 $L_{2}$ 点附近与月球以相同的周期 $T_{0}$ 一起绕地球做匀速圆周运动; 二是在与地月连线垂直的平面内绕 $L_{2}$ 点做匀速圆周运动. 已知地球的质量为月球质量的 $n$ 倍, 地球到 $L_{2}$ 点的距离为月球到 $L_{2}$ 点的距离的 $k$ 倍, 地球半径、月球半径以及“鹊桥”绕 $L_{2}$ 点做匀速圆周运动的半径均远小于月球到 $L_{2}$ 点的距离 (提示: “鹊桥”绕 $L_{2}$ 点做匀速圆周运动的向心力由地球和月球对其引力在过 $L_{2}$ 点与地月连线垂直的平面内的分量提供).[图1] 试推导“鹊桥”绕 $L_{2}$ 点做匀速圆周运动的周期 $T$ 的（近似）表达式. 若 $k=7, n=81$, $T_{0}=27.3$ 天, 求出 $T$ 的天数（取整数）. 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，你的答案应以天为单位计算，但在给出最终答案时，请不要包含单位。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。

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Astronomy

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multi-modal

Astronomy_136

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: 卫星 a 离地面的高度 $h$ 。 [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: 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: 卫星 a 离地面的高度 $h$ 。 [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_1213

The surface of the Sun has a temperature of $\sim 5700 \mathrm{~K}$ yet the solar corona (a very faint region of plasma normally only visible from Earth during a solar eclipse) is considerably hotter at around $10^{6} \mathrm{~K}$. The source of coronal heating is a mystery and so understanding how this might happen is one of several key science objectives of the Solar Orbiter spacecraft. It is equipped with an array of cameras and will take photos of the Sun from distances closer than ever before (other probes will go closer, but none of those have cameras). [figure1] Figure 7: Left: The Sun's corona (coloured green) as viewed in visible light (580-640 nm) taken with the METIS coronagraph instrument onboard Solar Orbiter. The coronagraph is a disc that blocks out the light of the Sun (whose size and position is indicated with the white circle in the middle) so that the faint corona can be seen. This was taken just after first perihelion and is already at a resolution only matched by ground-based telescopes during a solar eclipse - once it gets into the main phase of the mission when it is even closer then its photos will be unrivalled. Credit: METIS Team / ESA \& NASA Right: A high-resolution image from the Extreme Ultraviolet Imager (EUI), taken with the $\mathrm{HRI}_{\mathrm{EUV}}$ telescope just before first perihelion. The circle in the lower right corner indicates the size of Earth for scale. The arrow points to one of the ubiquitous features of the solar surface, called 'campfires', that were discovered by this spacecraft and may play an important role in heating the corona. Credit: EUI Team / ESA \& NASA. Launched in February 2020 (and taken to be at aphelion at launch), it arrived at its first perihelion on $15^{\text {th }}$ June 2020 and has sent back some of the highest resolution images of the surface of the Sun (i.e. the base of the corona) we have ever seen. In them we have identified phenomena nicknamed as 'campfires' (see Fig 7) which are already being considered as a potential major contributor to the mechanism of coronal heating. Later on in its mission it will go in even closer, and so will take photos of the Sun in unprecedented detail. The highest resolution photos are taken with the Extreme Ultraviolet Imager (EUI), which consists of three separate cameras. One of them, the Extreme Ultraviolet High Resolution Imager (HRI $\mathrm{HUV}$ ), is designed to pick up an emission line from highly ionised atoms of iron in the corona. The iron being detected has lost 9 electrons (i.e. $\mathrm{Fe}^{9+}$ ) though is called $\mathrm{Fe} \mathrm{X} \mathrm{('ten')} \mathrm{by} \mathrm{astronomers} \mathrm{(as} \mathrm{Fe} \mathrm{I} \mathrm{is} \mathrm{the} \mathrm{neutral}$ atom). Its presence can be used to work out the temperature of the part of the corona being investigated by the instrument. The photons detected by $\mathrm{HRI}_{\mathrm{EUV}}$ are emitted by a rearrangement of the electrons in the $\mathrm{Fe} \mathrm{X}$ ion, corresponding to a photon energy of $71.0372 \mathrm{eV}$ (where $1 \mathrm{eV}=1.60 \times 10^{-19} \mathrm{~J}$ ). The HRI $\mathrm{HUV}_{\mathrm{EUV}}$ telescope has a $1000^{\prime \prime}$ by $1000^{\prime \prime}$ field of view (FOV, where $1^{\circ}=3600^{\prime \prime}=3600$ arcseconds), an entrance pupil diameter of $47.4 \mathrm{~mm}$, a couple of mirrors that give an effective focal length of $4187 \mathrm{~mm}$, and the image is captured by a CCD with 2048 by 2048 pixels, each of which is 10 by $10 \mu \mathrm{m}$. Although we are viewing the emissions of $\mathrm{Fe} \mathrm{X}$ ions, the vast majority of the plasma in the corona is hydrogen and helium, and the bulk motions of this determine the timescales over which visible phenomena change. In particular, the speed of sound is very important if we do not want motion blur to affect our high resolution images, as this sets the limit on exposure times.c. The Rayleigh criterion and speed of sound in a plasma are given. iii. Later on in its mission, Solar Orbiter will have a perihelion of 0.284 au. Calculate the physical size on the Sun (in $\mathrm{km}$ ) of each picture element in an image taken with $\mathrm{HRI}_{\text {EUv }}$ as well as the FOV in units of $R_{\odot}$.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is an expression. Here is some context information for this question, which might assist you in solving it: The surface of the Sun has a temperature of $\sim 5700 \mathrm{~K}$ yet the solar corona (a very faint region of plasma normally only visible from Earth during a solar eclipse) is considerably hotter at around $10^{6} \mathrm{~K}$. The source of coronal heating is a mystery and so understanding how this might happen is one of several key science objectives of the Solar Orbiter spacecraft. It is equipped with an array of cameras and will take photos of the Sun from distances closer than ever before (other probes will go closer, but none of those have cameras). [figure1] Figure 7: Left: The Sun's corona (coloured green) as viewed in visible light (580-640 nm) taken with the METIS coronagraph instrument onboard Solar Orbiter. The coronagraph is a disc that blocks out the light of the Sun (whose size and position is indicated with the white circle in the middle) so that the faint corona can be seen. This was taken just after first perihelion and is already at a resolution only matched by ground-based telescopes during a solar eclipse - once it gets into the main phase of the mission when it is even closer then its photos will be unrivalled. Credit: METIS Team / ESA \& NASA Right: A high-resolution image from the Extreme Ultraviolet Imager (EUI), taken with the $\mathrm{HRI}_{\mathrm{EUV}}$ telescope just before first perihelion. The circle in the lower right corner indicates the size of Earth for scale. The arrow points to one of the ubiquitous features of the solar surface, called 'campfires', that were discovered by this spacecraft and may play an important role in heating the corona. Credit: EUI Team / ESA \& NASA. Launched in February 2020 (and taken to be at aphelion at launch), it arrived at its first perihelion on $15^{\text {th }}$ June 2020 and has sent back some of the highest resolution images of the surface of the Sun (i.e. the base of the corona) we have ever seen. In them we have identified phenomena nicknamed as 'campfires' (see Fig 7) which are already being considered as a potential major contributor to the mechanism of coronal heating. Later on in its mission it will go in even closer, and so will take photos of the Sun in unprecedented detail. The highest resolution photos are taken with the Extreme Ultraviolet Imager (EUI), which consists of three separate cameras. One of them, the Extreme Ultraviolet High Resolution Imager (HRI $\mathrm{HUV}$ ), is designed to pick up an emission line from highly ionised atoms of iron in the corona. The iron being detected has lost 9 electrons (i.e. $\mathrm{Fe}^{9+}$ ) though is called $\mathrm{Fe} \mathrm{X} \mathrm{('ten')} \mathrm{by} \mathrm{astronomers} \mathrm{(as} \mathrm{Fe} \mathrm{I} \mathrm{is} \mathrm{the} \mathrm{neutral}$ atom). Its presence can be used to work out the temperature of the part of the corona being investigated by the instrument. The photons detected by $\mathrm{HRI}_{\mathrm{EUV}}$ are emitted by a rearrangement of the electrons in the $\mathrm{Fe} \mathrm{X}$ ion, corresponding to a photon energy of $71.0372 \mathrm{eV}$ (where $1 \mathrm{eV}=1.60 \times 10^{-19} \mathrm{~J}$ ). The HRI $\mathrm{HUV}_{\mathrm{EUV}}$ telescope has a $1000^{\prime \prime}$ by $1000^{\prime \prime}$ field of view (FOV, where $1^{\circ}=3600^{\prime \prime}=3600$ arcseconds), an entrance pupil diameter of $47.4 \mathrm{~mm}$, a couple of mirrors that give an effective focal length of $4187 \mathrm{~mm}$, and the image is captured by a CCD with 2048 by 2048 pixels, each of which is 10 by $10 \mu \mathrm{m}$. Although we are viewing the emissions of $\mathrm{Fe} \mathrm{X}$ ions, the vast majority of the plasma in the corona is hydrogen and helium, and the bulk motions of this determine the timescales over which visible phenomena change. In particular, the speed of sound is very important if we do not want motion blur to affect our high resolution images, as this sets the limit on exposure times. problem: c. The Rayleigh criterion and speed of sound in a plasma are given. iii. Later on in its mission, Solar Orbiter will have a perihelion of 0.284 au. Calculate the physical size on the Sun (in $\mathrm{km}$ ) of each picture element in an image taken with $\mathrm{HRI}_{\text {EUv }}$ as well as the FOV in units of $R_{\odot}$. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is an expression without equals signs, e.g. ANSWER=\frac{1}{2} g t^2

EX

Astronomy

EN

multi-modal

Astronomy_231

卫星 1 和卫星 2 分别沿圆轨道和椭圆轨道环绕地球运行, 两轨道在同一平面内相交于 $A 、 B$ 两点, 卫星 2 在近地点离地心的距离是卫星 1 轨道半径的 $\frac{1}{k}$ 倍, 如图所示, 某时刻两卫星与地心在同一直线上, $D$ 点为远地点, 当卫星 2 运行到 $A$ 点时速度方向与 $C D$连线平行。已知近地点的曲率半径为 $\rho=\frac{b^{2}}{a}$, 式中 $a 、 b$ 分别是椭圆的半长轴和半短轴,下列说法中正确的是（） [图1] A: 两卫星在 $A$ 点的加速度相同 B: 卫星 2 在近地点的加速度大小是卫星 1 加速度大小的 $k$ 倍 C: 卫星 2 在近地点的速率是卫星 1 速率的 $k$ 倍 D: 卫星 2 运行到近地点时, 卫星 1 和卫星 2 的连线过地心

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: 卫星 1 和卫星 2 分别沿圆轨道和椭圆轨道环绕地球运行, 两轨道在同一平面内相交于 $A 、 B$ 两点, 卫星 2 在近地点离地心的距离是卫星 1 轨道半径的 $\frac{1}{k}$ 倍, 如图所示, 某时刻两卫星与地心在同一直线上, $D$ 点为远地点, 当卫星 2 运行到 $A$ 点时速度方向与 $C D$连线平行。已知近地点的曲率半径为 $\rho=\frac{b^{2}}{a}$, 式中 $a 、 b$ 分别是椭圆的半长轴和半短轴,下列说法中正确的是（） [图1] A: 两卫星在 $A$ 点的加速度相同 B: 卫星 2 在近地点的加速度大小是卫星 1 加速度大小的 $k$ 倍 C: 卫星 2 在近地点的速率是卫星 1 速率的 $k$ 倍 D: 卫星 2 运行到近地点时, 卫星 1 和卫星 2 的连线过地心 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在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_983

The very first image released by the James Webb Space Telescope (JWST) was of a galaxy cluster called SMACS 0723. The image is considered to be Webb's first deep field, since a long exposure time of 12.5 hours was used to allow the light from very faint and distant galaxies to be seen. The spectrum of one such galaxy is shown in Figure 2. [figure1] Figure 2: Highly redshifted emission lines in the spectrum of a galaxy that is 13.1 billion years old, captured using the JWST's near-infrared spectrometer (NIRSpec). Credit: NASA, ESA, CSA, STScI. The spectrum shows four bright hydrogen lines, which are part of the Balmer series (some of which are normally seen in the visible). The rest frame wavelengths of the longest four lines in the series are $410 \mathrm{~nm}, 434 \mathrm{~nm}, 486 \mathrm{~nm}$ and $656 \mathrm{~nm}$ (not all of which are visible in the spectrum). Once a redshift is known, its recessional velocity can be calculated. At very high redshifts, such as these, General Relativity must be used. A conversion from redshift to recessional velocity is shown in Figure 3. [figure2] Figure 3: Conversion from redshift to recessional velocity for a linear approximation $(v=z c)$, using Special Relativity $\left(v=c \frac{(1+z)^{2}-1}{(1+z)^{2}+1}\right)$, and using General Relativity $\left(v=\dot{a}(z) \int_{0}^{z} \frac{c d z^{\prime}}{H\left(z^{\prime}\right)}\right)$. The grey area corresponds to a variety of values for cosmological parameters. The solid line corresponds to values approximately the same as the current measured cosmological parameters. Credit: Davis \& Lineweaver (2001). Taking the value of the Hubble constant to be $H_{0}=70 \mathrm{~km} \mathrm{~s}^{-1} \mathrm{Mpc}^{-1}$, what is the distance to the galaxy? Give your answer in Mpc.

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 very first image released by the James Webb Space Telescope (JWST) was of a galaxy cluster called SMACS 0723. The image is considered to be Webb's first deep field, since a long exposure time of 12.5 hours was used to allow the light from very faint and distant galaxies to be seen. The spectrum of one such galaxy is shown in Figure 2. [figure1] Figure 2: Highly redshifted emission lines in the spectrum of a galaxy that is 13.1 billion years old, captured using the JWST's near-infrared spectrometer (NIRSpec). Credit: NASA, ESA, CSA, STScI. The spectrum shows four bright hydrogen lines, which are part of the Balmer series (some of which are normally seen in the visible). The rest frame wavelengths of the longest four lines in the series are $410 \mathrm{~nm}, 434 \mathrm{~nm}, 486 \mathrm{~nm}$ and $656 \mathrm{~nm}$ (not all of which are visible in the spectrum). Once a redshift is known, its recessional velocity can be calculated. At very high redshifts, such as these, General Relativity must be used. A conversion from redshift to recessional velocity is shown in Figure 3. [figure2] Figure 3: Conversion from redshift to recessional velocity for a linear approximation $(v=z c)$, using Special Relativity $\left(v=c \frac{(1+z)^{2}-1}{(1+z)^{2}+1}\right)$, and using General Relativity $\left(v=\dot{a}(z) \int_{0}^{z} \frac{c d z^{\prime}}{H\left(z^{\prime}\right)}\right)$. The grey area corresponds to a variety of values for cosmological parameters. The solid line corresponds to values approximately the same as the current measured cosmological parameters. Credit: Davis \& Lineweaver (2001). Taking the value of the Hubble constant to be $H_{0}=70 \mathrm{~km} \mathrm{~s}^{-1} \mathrm{Mpc}^{-1}$, what is the distance to the galaxy? Give your answer in Mpc. 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 Mpc, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

multi-modal

Astronomy_1015

The image below was released in April 2019 by the Event Horizon Telescope collaboration and is considered to be one of the most significant astronomical images ever made. What is it of? [figure1] A: A supernova B: A planetary nebula C: A black hole D: A quasar

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 image below was released in April 2019 by the Event Horizon Telescope collaboration and is considered to be one of the most significant astronomical images ever made. What is it of? [figure1] A: A supernova B: A planetary nebula C: A black hole D: A quasar 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_349

在星球 $P$ 和星球 $Q$ 的表面, 以相同的初速度 $v_{0}$ 坚直上抛一小球, 小球在空中运动时的 $v-t$ 图像分别如图所示。假设两星球均为质量均匀分布的球体, 星球 $P$ 的半径是星球 $Q$ 半径的 3 倍, 下列说法正确的是（） [图1] A: 星球 $P$ 和星球 $Q$ 的质量之比为 $3: 1$ B: 星球 $P$ 和星球 $Q$ 的密度之比为 $1: 1$ C: 星球 $P$ 和星球 $Q$ 的第一宇宙速度之比为 3:1 D: 星球 $P$ 和星球 $Q$ 的近地卫星周期之比为 $1: 3$

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: 在星球 $P$ 和星球 $Q$ 的表面, 以相同的初速度 $v_{0}$ 坚直上抛一小球, 小球在空中运动时的 $v-t$ 图像分别如图所示。假设两星球均为质量均匀分布的球体, 星球 $P$ 的半径是星球 $Q$ 半径的 3 倍, 下列说法正确的是（） [图1] A: 星球 $P$ 和星球 $Q$ 的质量之比为 $3: 1$ B: 星球 $P$ 和星球 $Q$ 的密度之比为 $1: 1$ C: 星球 $P$ 和星球 $Q$ 的第一宇宙速度之比为 3:1 D: 星球 $P$ 和星球 $Q$ 的近地卫星周期之比为 $1: 3$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_822

The apparent magnitude of a star of radius $0.41 R_{\odot}$ as observed from Earth appears to fluctuate by 0.037 . That is, the difference between the maximum and minimum apparent magnitudes is 0.037 . This fluctuation is caused by an exoplanet that orbits the star. Determine the radius of the exoplanet. A: $0.075 R_{\odot}$ B: $0.079 R_{\odot}$ C: $0.085 R_{\odot}$ D: $0.098 R_{\odot}$ E: $0.12 R_{\odot}$

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 apparent magnitude of a star of radius $0.41 R_{\odot}$ as observed from Earth appears to fluctuate by 0.037 . That is, the difference between the maximum and minimum apparent magnitudes is 0.037 . This fluctuation is caused by an exoplanet that orbits the star. Determine the radius of the exoplanet. A: $0.075 R_{\odot}$ B: $0.079 R_{\odot}$ C: $0.085 R_{\odot}$ D: $0.098 R_{\odot}$ E: $0.12 R_{\odot}$ 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_924

In which constellation would you find the centre of the Milky Way? A: Ophiucus B: Coma Berenices C: Sagittarius D: Scorpius

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: In which constellation would you find the centre of the Milky Way? A: Ophiucus B: Coma Berenices C: Sagittarius D: Scorpius 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_230

$\mathrm{A} 、 \mathrm{~B}$ 两颗卫星在同一平面内沿同一方向绕地球做匀速圆周运动, 如图甲所示。两卫星之间的距离 $\Delta r$ 随时间周期性变化, 如图乙所示。仅考虑地球对卫星的引力, 下列说法正确的是（ ） [图1] 图(a) [图2] 图(b) A: A、B 的轨道半径之比为 $1: 3$ B: A、B 的线速度之比为 $1: 2$ C: $\mathrm{A}$ 的运动周期大于 $\mathrm{B}$ 的运动周期 D: $A 、 B$ 的向心加速度之比为 4: 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: $\mathrm{A} 、 \mathrm{~B}$ 两颗卫星在同一平面内沿同一方向绕地球做匀速圆周运动, 如图甲所示。两卫星之间的距离 $\Delta r$ 随时间周期性变化, 如图乙所示。仅考虑地球对卫星的引力, 下列说法正确的是（ ） [图1] 图(a) [图2] 图(b) A: A、B 的轨道半径之比为 $1: 3$ B: A、B 的线速度之比为 $1: 2$ C: $\mathrm{A}$ 的运动周期大于 $\mathrm{B}$ 的运动周期 D: $A 、 B$ 的向心加速度之比为 4: 1 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

multi-modal

Astronomy_208

宇宙空间由一种由三颗星体 $A 、 B 、 C$ 组成的三星体系, 它们分别位于等边三角形 $A B C$的三个顶点上, 绕一个固定且共同的圆心 $O$ 做匀速圆周运动, 轨道如图中实线所示,其轨道半径 $r_{A}<r_{B}<r_{C}$. 忽略其他星体对它们的作用, 关于这三颗星体, 下列说法正确的是 ( ) [图1] A: 线速度大小关系是 $v_{A}<v_{B}<v_{C}$ B: 加速度大小关系是 $a_{A}<a_{B}<a_{C}$ C: 质量大小关系是 $m_{A}=m_{B}=m_{C}$ D: 角速度大小关系是 $\omega_{A}=\omega_{B}=\omega_{C}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 宇宙空间由一种由三颗星体 $A 、 B 、 C$ 组成的三星体系, 它们分别位于等边三角形 $A B C$的三个顶点上, 绕一个固定且共同的圆心 $O$ 做匀速圆周运动, 轨道如图中实线所示,其轨道半径 $r_{A}<r_{B}<r_{C}$. 忽略其他星体对它们的作用, 关于这三颗星体, 下列说法正确的是 ( ) [图1] A: 线速度大小关系是 $v_{A}<v_{B}<v_{C}$ B: 加速度大小关系是 $a_{A}<a_{B}<a_{C}$ C: 质量大小关系是 $m_{A}=m_{B}=m_{C}$ D: 角速度大小关系是 $\omega_{A}=\omega_{B}=\omega_{C}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_342

2020 年 11 月 6 日, 我国成功发射全球首颗 6G 试验地球卫星, 卫星的轨道半径的三次方与其周期的二次方的关系图像如图所示。已知地球半径为 $R$, 引力常量为 $G$, 下列说法正确的是 ( ) [图1] A: 地球的质量为 $\frac{4 \pi^{2} a}{b G}$ B: 地球表面的重力加速度为 $\frac{4 \pi^{2} b}{a R^{2}}$ C: 绕地球表面运行的卫星的线速度大小为 $\sqrt{\frac{4 \pi^{2} b}{R a}}$ D: 地球密度为 $\frac{3 \pi a}{b G R^{3}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 2020 年 11 月 6 日, 我国成功发射全球首颗 6G 试验地球卫星, 卫星的轨道半径的三次方与其周期的二次方的关系图像如图所示。已知地球半径为 $R$, 引力常量为 $G$, 下列说法正确的是 ( ) [图1] A: 地球的质量为 $\frac{4 \pi^{2} a}{b G}$ B: 地球表面的重力加速度为 $\frac{4 \pi^{2} b}{a R^{2}}$ C: 绕地球表面运行的卫星的线速度大小为 $\sqrt{\frac{4 \pi^{2} b}{R a}}$ D: 地球密度为 $\frac{3 \pi a}{b G R^{3}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_850

A star with mass $M$ goes through an energy generating nuclear reaction $4{ }^{1} H \rightarrow{ }^{4} \mathrm{He}+$ Energy. Here, the burning efficiency of the $\mathrm{p}$-p(proton-proton) chain is 0.007 , meaning that each mass $m$ yields $0.007 m c^{2}$ of energy. Assuming that the has a total available hydrogen mass for nuclear reaction amounts to half of its original mass, and the luminosity $(L)$ stays constant throughout the burning phase, get an expression of the hydrogen burning lifetime of the star. A: $1.625 \times 10^{18} s\left(\frac{M}{M_{\odot}}\right)^{-2}$ B: $3.15 \times 10^{14} s\left(\frac{M}{M_{\odot}}\right)^{-2}$ C: $1.625 \times 10^{18} s\left(\frac{M}{M_{\odot}}\right)^{2}$ D: $3.15 \times 10^{14} s\left(\frac{M}{M_{\odot}}\right)^{2}$ E: None of the above

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: A star with mass $M$ goes through an energy generating nuclear reaction $4{ }^{1} H \rightarrow{ }^{4} \mathrm{He}+$ Energy. Here, the burning efficiency of the $\mathrm{p}$-p(proton-proton) chain is 0.007 , meaning that each mass $m$ yields $0.007 m c^{2}$ of energy. Assuming that the has a total available hydrogen mass for nuclear reaction amounts to half of its original mass, and the luminosity $(L)$ stays constant throughout the burning phase, get an expression of the hydrogen burning lifetime of the star. A: $1.625 \times 10^{18} s\left(\frac{M}{M_{\odot}}\right)^{-2}$ B: $3.15 \times 10^{14} s\left(\frac{M}{M_{\odot}}\right)^{-2}$ C: $1.625 \times 10^{18} s\left(\frac{M}{M_{\odot}}\right)^{2}$ D: $3.15 \times 10^{14} s\left(\frac{M}{M_{\odot}}\right)^{2}$ E: None of the above You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D, E].

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Astronomy

EN

text-only

Astronomy_279

如图, 地球和某行星在同一轨道平面内同向绕太阳做匀速圆周运动。地球的运转周期为 $T$ 。地球和太阳的连线与地球和行星的连线所夹的角叫地球对该行星的观察视角 (简称视角)。已知该行星的最大视角为 $\theta$, 当行星处于最大视角处时, 是地球上天文爱好者观察该行星的最佳时期。 求行星绕太阳转动的角速度与地球绕太阳转动的角速度之比; [图1]

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 如图, 地球和某行星在同一轨道平面内同向绕太阳做匀速圆周运动。地球的运转周期为 $T$ 。地球和太阳的连线与地球和行星的连线所夹的角叫地球对该行星的观察视角 (简称视角)。已知该行星的最大视角为 $\theta$, 当行星处于最大视角处时, 是地球上天文爱好者观察该行星的最佳时期。 求行星绕太阳转动的角速度与地球绕太阳转动的角速度之比; [图1] 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

EX

Astronomy

ZH

multi-modal

Astronomy_885

Consider a horizontal sundial where the triangular gnomon rises at an angle equal to the sundial site's latitude, $\phi=38^{\circ}$. If the area of the triangular gnomon is $2 \mathrm{~m}^{2}$, what would be the area of the shadow in $m^{2}$ three hours after the noon in the first day of spring (vernal equinox)? [figure1] A: 3.0 B: 3.5 C: 2.5 D: 1.5 E: 4.0

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: Consider a horizontal sundial where the triangular gnomon rises at an angle equal to the sundial site's latitude, $\phi=38^{\circ}$. If the area of the triangular gnomon is $2 \mathrm{~m}^{2}$, what would be the area of the shadow in $m^{2}$ three hours after the noon in the first day of spring (vernal equinox)? [figure1] A: 3.0 B: 3.5 C: 2.5 D: 1.5 E: 4.0 You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D, E].

SC

Astronomy

EN

multi-modal

Astronomy_486

如图所示, 曲线 $\mathrm{I}$ 是一颗绕地球做圆周运动的卫星 $P$ 轨道的示意图, 其半径为 $R$;曲线II是一颗绕地球做椭圆运动的卫星 $Q$ 轨道的示意图, $O$ 点为地球球心, $A B$ 为椭圆的长轴, 两轨道和地心都在同一平面内, 已知在两轨道上运动的卫星的周期相等, 万有引力常量为 $G$ ，地球质量为 $M$ ，下列说法错误的是（ ） [图1] A: 椭圆轨道的长轴长度为 $2 R$ B: 卫星 $P$ 在 I 轨道的速率为 $v_{0}$, 卫星 $Q$ 在II轨道 $B$ 点的速率为 $\mathrm{v}_{\mathrm{B}}$, 则 $v_{0}>\mathrm{v}_{\mathrm{B}}$ C: 卫星 $P$ 在 I 轨道的加速度大小为 $a_{0}$, 卫星 $Q$ 在II轨道 $A$ 点加速度大小为 $\mathrm{a}_{\mathrm{A}}$, 则 $a_{0}<\mathrm{a}_{\mathrm{A}}$ D: 卫星 $P$ 在 I 轨道上受到的地球引力与卫星 $Q$ 在II轨道上经过两轨道交点时受到的地球引力大小相等

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 如图所示, 曲线 $\mathrm{I}$ 是一颗绕地球做圆周运动的卫星 $P$ 轨道的示意图, 其半径为 $R$;曲线II是一颗绕地球做椭圆运动的卫星 $Q$ 轨道的示意图, $O$ 点为地球球心, $A B$ 为椭圆的长轴, 两轨道和地心都在同一平面内, 已知在两轨道上运动的卫星的周期相等, 万有引力常量为 $G$ ，地球质量为 $M$ ，下列说法错误的是（ ） [图1] A: 椭圆轨道的长轴长度为 $2 R$ B: 卫星 $P$ 在 I 轨道的速率为 $v_{0}$, 卫星 $Q$ 在II轨道 $B$ 点的速率为 $\mathrm{v}_{\mathrm{B}}$, 则 $v_{0}>\mathrm{v}_{\mathrm{B}}$ C: 卫星 $P$ 在 I 轨道的加速度大小为 $a_{0}$, 卫星 $Q$ 在II轨道 $A$ 点加速度大小为 $\mathrm{a}_{\mathrm{A}}$, 则 $a_{0}<\mathrm{a}_{\mathrm{A}}$ D: 卫星 $P$ 在 I 轨道上受到的地球引力与卫星 $Q$ 在II轨道上经过两轨道交点时受到的地球引力大小相等 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

multi-modal

Astronomy_1104

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

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is an equation. Here is some context information for this question, which might assist you in solving it: The Parker Solar Probe (PSP) is part of a mission to learn more about the Sun, named after the scientist that first proposed the existence of the solar wind, and was launched on $12^{\text {th }}$ August 2018. Over the course of the 7 year mission it will orbit the Sun 24 times, and through 7 flybys of Venus it will lose some energy in order to get into an ever tighter orbit (see Figure 1). In its final 3 orbits it will have a perihelion (closest approach to the Sun) of only $r_{\text {peri }}=9.86 R_{\odot}$, about 7 times closer than any previous probe, the first of which is due on $24^{\text {th }}$ December 2024. In this extreme environment the probe will not only face extreme brightness and temperatures but also will break the record for the fastest ever spacecraft. [figure1] Figure 1: Left: The journey PSP will take to get from the Earth to the final orbit around the Sun. Right: The probe just after assembly in the John Hopkins University Applied Physics Laboratory. Credit: NASA / John Hopkins APL / Ed Whitman. $$ v^{2}=G M\left(\frac{2}{r}-\frac{1}{a}\right) $$ Given that in its final orbit PSP has a orbital period of 88 days, calculate the speed of the probe as it passes through the minimum perihelion. Give your answer in $\mathrm{km} \mathrm{s}^{-1}$. Close to the Sun the communications equipment is very sensitive to the extreme environment, so the mission is planned for the probe to take all of its primary science measurements whilst within 0.25 au of the Sun, and then to spend the rest of the orbit beaming that data back to Earth, as shown in Figure 2. [figure2] Figure 2: The way PSP is planned to split each orbit into taking measurements and sending data back. Credit: NASA / Johns Hopkins APL. When considering the position of an object in an elliptical orbit as a function of time, there are two important angles (called 'anomalies') necessary to do the calculation, and they are defined in Figure 3. By constructing a circular orbit centred on the same point as the ellipse and with the same orbital period, the eccentric anomaly, $E$, is then the angle between the major axis and the perpendicular projection of the object (some time $t$ after perihelion) onto the circle as measured from the centre of the ellipse ( $\angle x c z$ in the figure). The mean anomaly, $M$, is the angle between the major axis and where the object would have been at time $t$ if it was indeed on the circular orbit ( $\angle y c z$ in the figure, such that the shaded areas are the same). [figure3] Figure 3: The definitions of the anomalies needed to get the position of an object in an ellipse as a function of time. The Sun (located at the focus) is labeled $S$ and the probe $P . M$ and $E$ are the mean and eccentric anomalies respectively. The angle $\theta$ is called the true anomaly and is not needed for this question. Credit: Wikipedia. The eccentric anomaly can be related to the mean anomaly through Kepler's Equation, $$ M=E-e \sin E \text {. } $$ problem: d. Derive a formula for the distance from the focus for an elliptical orbit, $r$ (SP in the figure) in terms of the semi-major axis $a$, the eccentricity $e$, and the eccentric anomaly $E$. 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 equation, e.g. ANSWER=\frac{x^2}{4}+\frac{y^2}{2}=1

EQ

Astronomy

EN

multi-modal

Astronomy_838

An exoplanet was observed during its transit across the surface of a bright star. Estimate the variation of the apparent magnitude $(\Delta \mathrm{m})$ of the star caused by exoplanet's transit. During the transit, assume an Earth-based astronomer observes that the area covered by the exoplanet on the projected surface of the star represents $\eta=2 \%$ of the star's projected surface. A: -4.247 B: 0.003 C: 0.022 D: 0.679 E: -0.003

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: An exoplanet was observed during its transit across the surface of a bright star. Estimate the variation of the apparent magnitude $(\Delta \mathrm{m})$ of the star caused by exoplanet's transit. During the transit, assume an Earth-based astronomer observes that the area covered by the exoplanet on the projected surface of the star represents $\eta=2 \%$ of the star's projected surface. A: -4.247 B: 0.003 C: 0.022 D: 0.679 E: -0.003 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_396

有一颗绕地球做匀速圆周运动的卫星, 其运行的线速度是地球近地卫星的 $\frac{\sqrt{2}}{2}$, 卫星圆形轨道平面与地球赤道平面重合, 卫星上有太阳能收集板可以把光能转化为电能,提供卫星工作所必须的能量。已知地球表面重力加速度为 $g$, 地球半径为 $R$, 忽略地球公转, 此时太阳处于赤道平面上, 近似认为太阳光是平行光, 则下列说法正确的是 ( ) A: 卫星的轨道半径为 $3 R$ B: 卫星轨道所在位置的重力加速度为 $\frac{1}{3} g$ C: 卫星运动的周期为 $4 \pi \sqrt{\frac{2 R}{g}}$ D: 卫星绕地球一周, 太阳能收集板的工作时间为 $\frac{10 \pi}{3} \sqrt{\frac{2 R}{g}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 有一颗绕地球做匀速圆周运动的卫星, 其运行的线速度是地球近地卫星的 $\frac{\sqrt{2}}{2}$, 卫星圆形轨道平面与地球赤道平面重合, 卫星上有太阳能收集板可以把光能转化为电能,提供卫星工作所必须的能量。已知地球表面重力加速度为 $g$, 地球半径为 $R$, 忽略地球公转, 此时太阳处于赤道平面上, 近似认为太阳光是平行光, 则下列说法正确的是 ( ) A: 卫星的轨道半径为 $3 R$ B: 卫星轨道所在位置的重力加速度为 $\frac{1}{3} g$ C: 卫星运动的周期为 $4 \pi \sqrt{\frac{2 R}{g}}$ D: 卫星绕地球一周, 太阳能收集板的工作时间为 $\frac{10 \pi}{3} \sqrt{\frac{2 R}{g}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

text-only

Astronomy_519

已知某卫星在赤道上空的圆形轨道运行, 轨道半径为 $r_{1}$, 运行周期为 $T$, 卫星运动方向与地球自转方向相同, 不计空气阻力, 万有引力常量为 $G \circ$ 。求: 地球质量 $M$ 的大小; [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: 已知某卫星在赤道上空的圆形轨道运行, 轨道半径为 $r_{1}$, 运行周期为 $T$, 卫星运动方向与地球自转方向相同, 不计空气阻力, 万有引力常量为 $G \circ$ 。求: 地球质量 $M$ 的大小; [图1] 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

EX

Astronomy

ZH

multi-modal

Astronomy_137

有人设想: 可以在飞船从运行轨道进入返回地球程序时, 借飞船需要减速的机会,发射一个小型太空探测器, 从而达到节能的目的。如图所示, 飞船在圆轨道I上绕地球飞行, 其轨道半径为地球半径的 $k$ 倍 $(k>1)$ 。当飞船通过轨道I的 $A$ 点时, 飞船上的发射装置短暂工作, 将探测器沿飞船原运动方向射出, 并使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, 而飞船在发射探测器后沿椭圆轨道II向前运动, 其近地点 $B$ 到地心的距离近似为地球半径 $R$ 。已知取无穷远处引力势能为零,物体距星球球心距离为 $r$ 时的引力势能 $E_{\mathrm{p}}=-G \frac{M m}{r}$ 。在飞船沿轨道I和轨道II以及探测器被射出后的运动过程中, 其动能和引力势能之和均保持不变。以上过程中飞船和探测器的质量均可视为不变, 已知地球表面的重力加速度为 $g$ 。则下列说法正确的是 $(\quad)$ [图1] A: 飞船在轨道 $\mathrm{I}$ 运动的速度大小为 $\sqrt{(k+1) g R}$ B: 飞船在轨道I上的运行周期是在轨道II上运行周期的 $\frac{2 k}{k+1}$ 倍 C: 探测器刚离开飞船时的速度大小为 $\sqrt{\frac{2 g R}{k}}$ D: 若飞船沿轨道II运动过程中, 通过 $A$ 点与 $B$ 点的速度大小与这两点到地心的距离成反比, 实现上述飞船和探测器的运动过程, 飞船与探测器的质量之比应满足 $\frac{\sqrt{2}}{1-\sqrt{\frac{2}{k+1}}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 有人设想: 可以在飞船从运行轨道进入返回地球程序时, 借飞船需要减速的机会,发射一个小型太空探测器, 从而达到节能的目的。如图所示, 飞船在圆轨道I上绕地球飞行, 其轨道半径为地球半径的 $k$ 倍 $(k>1)$ 。当飞船通过轨道I的 $A$ 点时, 飞船上的发射装置短暂工作, 将探测器沿飞船原运动方向射出, 并使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, 而飞船在发射探测器后沿椭圆轨道II向前运动, 其近地点 $B$ 到地心的距离近似为地球半径 $R$ 。已知取无穷远处引力势能为零,物体距星球球心距离为 $r$ 时的引力势能 $E_{\mathrm{p}}=-G \frac{M m}{r}$ 。在飞船沿轨道I和轨道II以及探测器被射出后的运动过程中, 其动能和引力势能之和均保持不变。以上过程中飞船和探测器的质量均可视为不变, 已知地球表面的重力加速度为 $g$ 。则下列说法正确的是 $(\quad)$ [图1] A: 飞船在轨道 $\mathrm{I}$ 运动的速度大小为 $\sqrt{(k+1) g R}$ B: 飞船在轨道I上的运行周期是在轨道II上运行周期的 $\frac{2 k}{k+1}$ 倍 C: 探测器刚离开飞船时的速度大小为 $\sqrt{\frac{2 g R}{k}}$ D: 若飞船沿轨道II运动过程中, 通过 $A$ 点与 $B$ 点的速度大小与这两点到地心的距离成反比, 实现上述飞船和探测器的运动过程, 飞船与探测器的质量之比应满足 $\frac{\sqrt{2}}{1-\sqrt{\frac{2}{k+1}}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

multi-modal

Astronomy_1211

On $21^{\text {st }}$ December 2020, Jupiter and Saturn formed a true spectacle in the southwestern sky just after sunset in the UK, separated by only $0.102^{\circ}$. This is close enough that when viewed through a telescope both planets and their moons could be seen in the same field of view (see Fig 4). When two planets occupy the same piece of sky it is known as a conjunction, and when it is Jupiter and Saturn it is known as a great conjunction (so named because they are the rarest of the naked-eye planet conjunctions). The reason it happens is because Jupiter's orbital velocity is higher than Saturn's and so as time goes on Jupiter catches up with and overtakes Saturn (at least as viewed from Earth), with the moment of overtaking corresponding to the conjunction. The process of the two getting closer and closer together has been seen in sky throughout the year (see Fig 5). [figure1] Figure 4: Left: The view of the planets the day before their closest approach, as captured by the 16 " telescope at the Institute of Astronomy, Cambridge. Credit: Robin Catchpole. Right: The view from Arizona on the day of the closest approach as viewed with the Lowell Discovery Telescope. Credit: Levine / Elbert / Bosh / Lowell Observatory. [figure2] Figure 5: Left: Demonstrating how Jupiter and Saturn have been getting closer and closer together over the last few months. Separations are given in degrees and arcminutes $\left(1 / 60^{\text {th }}\right.$ of a degree). Credit: Pete Lawrence. Right: The positions of the Earth, Jupiter and Saturn that were responsible for the 2020 great conjunction. The precise timing of the apparent alignment is clearly sensitive to where Earth is in its orbit. Credit: timeanddate.com. The time between conjunctions is known as the synodic period. Although this period will change slightly from conjunction to conjunction due to different lines of perspective as viewed from Earth (see Fig 5), we can work out the average time between great conjunctions by considering both planets travelling on circular coplanar orbits and ignoring the position of the Earth. For circular coplanar orbits the centre of Jupiter's disc would pass in front of the centre of Saturn's disc every conjunction, and hence have an angular separation of $\theta=0^{\circ}$ (it is measured from the centre of each disc). In practice, the planets follow elliptical orbits that are in planes inclined at different angles to each other. Fig 6 shows how this affects the real values for over 8000 years' worth of data, which along with different synodic periods between conjunctions makes it a difficult problem to solve precisely without a computer. However, after one synodic period Saturn has moved about $2 / 3$ of the way around its orbit, and so roughly every 3 synodic periods it is in a similar part of the sky. Consequently every third great conjunction follows a reasonably regular pattern which can be fit with a sinusoidal function. [figure3] Figure 6: Top: All great conjunctions from 1800 to 2300, calculated for the real celestial mechanics of the Solar System. We can see that each great conjunction belongs to one of three different series or tracks, with Track A indicated with orange circles, Track B with green squares, and Track C with blue triangles. Bottom: The same idea but extended over a much larger date range, up to $10000 \mathrm{AD}$. It is clear the distinct series form broadly sinusoidal patterns which can be used with the average synodic period to give rough predictions for the separations of great conjunctions. The opacity of points is related to each conjunction's angular separation from the Sun (where low opacity means close to the Sun, so it is harder for any observers to see). Credit: Nick Koukoufilippas, but inspired by the work of Steffen Thorsen and Graham Jones / Sky \& Telescope.c. By empirically fitting a sinusoidal function (which is assumed to be the same for each track, just with a fixed phase difference between them) and assuming all conjunctions are separated by the average synodic period, we can give rough estimations for the separations of any given great conjunction. Note: be careful as your calculations will be very sensitive to rounding errors. i. By reading off the graph, give an equation for Track $A$ of the form $\theta=\left|D \sin \left(\frac{2 \pi t}{\lambda}+\phi_{A}\right)\right|$, where $t$ is the (decimalised) date in years, and $D, \lambda$, and $-\pi / 2<\phi_{A} \leq \pi / 2$ are values that need to be determined. [Hint: ensure your function passes through the 2020 data point, and the function is decreasing as it does.]

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: On $21^{\text {st }}$ December 2020, Jupiter and Saturn formed a true spectacle in the southwestern sky just after sunset in the UK, separated by only $0.102^{\circ}$. This is close enough that when viewed through a telescope both planets and their moons could be seen in the same field of view (see Fig 4). When two planets occupy the same piece of sky it is known as a conjunction, and when it is Jupiter and Saturn it is known as a great conjunction (so named because they are the rarest of the naked-eye planet conjunctions). The reason it happens is because Jupiter's orbital velocity is higher than Saturn's and so as time goes on Jupiter catches up with and overtakes Saturn (at least as viewed from Earth), with the moment of overtaking corresponding to the conjunction. The process of the two getting closer and closer together has been seen in sky throughout the year (see Fig 5). [figure1] Figure 4: Left: The view of the planets the day before their closest approach, as captured by the 16 " telescope at the Institute of Astronomy, Cambridge. Credit: Robin Catchpole. Right: The view from Arizona on the day of the closest approach as viewed with the Lowell Discovery Telescope. Credit: Levine / Elbert / Bosh / Lowell Observatory. [figure2] Figure 5: Left: Demonstrating how Jupiter and Saturn have been getting closer and closer together over the last few months. Separations are given in degrees and arcminutes $\left(1 / 60^{\text {th }}\right.$ of a degree). Credit: Pete Lawrence. Right: The positions of the Earth, Jupiter and Saturn that were responsible for the 2020 great conjunction. The precise timing of the apparent alignment is clearly sensitive to where Earth is in its orbit. Credit: timeanddate.com. The time between conjunctions is known as the synodic period. Although this period will change slightly from conjunction to conjunction due to different lines of perspective as viewed from Earth (see Fig 5), we can work out the average time between great conjunctions by considering both planets travelling on circular coplanar orbits and ignoring the position of the Earth. For circular coplanar orbits the centre of Jupiter's disc would pass in front of the centre of Saturn's disc every conjunction, and hence have an angular separation of $\theta=0^{\circ}$ (it is measured from the centre of each disc). In practice, the planets follow elliptical orbits that are in planes inclined at different angles to each other. Fig 6 shows how this affects the real values for over 8000 years' worth of data, which along with different synodic periods between conjunctions makes it a difficult problem to solve precisely without a computer. However, after one synodic period Saturn has moved about $2 / 3$ of the way around its orbit, and so roughly every 3 synodic periods it is in a similar part of the sky. Consequently every third great conjunction follows a reasonably regular pattern which can be fit with a sinusoidal function. [figure3] Figure 6: Top: All great conjunctions from 1800 to 2300, calculated for the real celestial mechanics of the Solar System. We can see that each great conjunction belongs to one of three different series or tracks, with Track A indicated with orange circles, Track B with green squares, and Track C with blue triangles. Bottom: The same idea but extended over a much larger date range, up to $10000 \mathrm{AD}$. It is clear the distinct series form broadly sinusoidal patterns which can be used with the average synodic period to give rough predictions for the separations of great conjunctions. The opacity of points is related to each conjunction's angular separation from the Sun (where low opacity means close to the Sun, so it is harder for any observers to see). Credit: Nick Koukoufilippas, but inspired by the work of Steffen Thorsen and Graham Jones / Sky \& Telescope. problem: c. By empirically fitting a sinusoidal function (which is assumed to be the same for each track, just with a fixed phase difference between them) and assuming all conjunctions are separated by the average synodic period, we can give rough estimations for the separations of any given great conjunction. Note: be careful as your calculations will be very sensitive to rounding errors. i. By reading off the graph, give an equation for Track $A$ of the form $\theta=\left|D \sin \left(\frac{2 \pi t}{\lambda}+\phi_{A}\right)\right|$, where $t$ is the (decimalised) date in years, and $D, \lambda$, and $-\pi / 2<\phi_{A} \leq \pi / 2$ are values that need to be determined. [Hint: ensure your function passes through the 2020 data point, and the function is decreasing as it does.] All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of ^{\circ}, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

multi-modal

Astronomy_592

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 $\left(\sqrt[3]{\frac{g_{0}}{g_{0}-g}}-1\right) R$ B: 地球的质量为 $\frac{g R^{2}}{G}$ C: 地球的第一宇宙速度为 $\sqrt{g_{0} R}$ D: 地球密度为 $\frac{3 g_{0}}{4 \pi R G}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 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 $\left(\sqrt[3]{\frac{g_{0}}{g_{0}-g}}-1\right) R$ B: 地球的质量为 $\frac{g R^{2}}{G}$ C: 地球的第一宇宙速度为 $\sqrt{g_{0} R}$ D: 地球密度为 $\frac{3 g_{0}}{4 \pi R G}$ 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应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

text-only

Astronomy_773

Kepler's Laws state that the cube of a planet's semi-major axis is proportional to the ... A: square of the average distance. B: covered area by the planet. C: cube of the average distance. D: square of the orbital period.

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: Kepler's Laws state that the cube of a planet's semi-major axis is proportional to the ... A: square of the average distance. B: covered area by the planet. C: cube of the average distance. D: square of the orbital period. 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_505

万有引力定律能够很好地将天体运行规律与地球上物体运动规律具有的内在一致性统一起来. 用弹簧秤称量一个相对于地球静止的小物体的重量, 随称量位置的变化可能会有不同的结果. 已知地球质量为 $M$, 万有引力常量为 $G$. 将地球视为半径为 $\mathrm{R}$ 质量均匀分布的球体. 下列选项中说法正确的是 A: 在赤道地面称量时, 弹簧秤读数为 $F_{1}=G \frac{M m}{R^{2}}$ B: 在北极地面称量时, 弹簧科读数为 $F_{0}=G \frac{M m}{R^{2}}$ C: 在北极上空高出地面 $h$ 处称量时, 弹簧秤读数为 $F_{2}=G \frac{M m}{(R+h)^{2}}$ D: 在赤道上空高出地面 $h$ 处称量时, 弹簧科读数为 $F_{3}=G \frac{M m}{(R+h)^{2}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 万有引力定律能够很好地将天体运行规律与地球上物体运动规律具有的内在一致性统一起来. 用弹簧秤称量一个相对于地球静止的小物体的重量, 随称量位置的变化可能会有不同的结果. 已知地球质量为 $M$, 万有引力常量为 $G$. 将地球视为半径为 $\mathrm{R}$ 质量均匀分布的球体. 下列选项中说法正确的是 A: 在赤道地面称量时, 弹簧秤读数为 $F_{1}=G \frac{M m}{R^{2}}$ B: 在北极地面称量时, 弹簧科读数为 $F_{0}=G \frac{M m}{R^{2}}$ C: 在北极上空高出地面 $h$ 处称量时, 弹簧秤读数为 $F_{2}=G \frac{M m}{(R+h)^{2}}$ D: 在赤道上空高出地面 $h$ 处称量时, 弹簧科读数为 $F_{3}=G \frac{M m}{(R+h)^{2}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

text-only

Astronomy_1113

The Sun is seen setting from London $\left(\varphi=51^{\circ} 30^{\prime} \mathrm{N}, L=0^{\circ} 8^{\prime} \mathrm{W}\right)$ at 21:00 UT. At what time UT will it be seen setting in Cardiff ( $\varphi=51^{\circ} 30^{\prime} \mathrm{N}, L=3^{\circ} 11^{\prime} \mathrm{W}$ ) on the same day? A: $21: 12$ B: $21: 00$ C: $20: 48$ D: $20: 58$

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 Sun is seen setting from London $\left(\varphi=51^{\circ} 30^{\prime} \mathrm{N}, L=0^{\circ} 8^{\prime} \mathrm{W}\right)$ at 21:00 UT. At what time UT will it be seen setting in Cardiff ( $\varphi=51^{\circ} 30^{\prime} \mathrm{N}, L=3^{\circ} 11^{\prime} \mathrm{W}$ ) on the same day? A: $21: 12$ B: $21: 00$ C: $20: 48$ D: $20: 58$ 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_710

现有质量相等的 $\mathrm{A} 、 \mathrm{~B} 、 \mathrm{C}$ 三个物体, 物体 $\mathrm{A}$ 置于地球表面赤道上随地球一起自转, $\mathrm{B}$ 是一颗近地轨道卫星, 在赤道正上方绕地球做匀速圆周运动（忽略其距离地面的高度), $\mathrm{C}$ 是一颗地球同步卫星。 $\mathrm{A} 、 \mathrm{~B} 、 \mathrm{C}$ 三个物体所受地球的万有引力大小分别为 $F A$ 、 $F B$ 和 $F C$; 它们绕地心做圆周运动的线速度大小分别为 $v A 、 v B$ 和 $v C$; 角速度大小分别为 $\omega A 、 \omega B$ 和 $\omega C$; 向心加速度大小分别为 $a A 、 a B$ 和 $a C$ 。则以下分析中正确的是 ( ) A: $F B>F C>F A$ B: $v B>v C>v A$ C: $\omega B>\omega C>\omega A$ D: $a B>a C>a A$

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: 现有质量相等的 $\mathrm{A} 、 \mathrm{~B} 、 \mathrm{C}$ 三个物体, 物体 $\mathrm{A}$ 置于地球表面赤道上随地球一起自转, $\mathrm{B}$ 是一颗近地轨道卫星, 在赤道正上方绕地球做匀速圆周运动（忽略其距离地面的高度), $\mathrm{C}$ 是一颗地球同步卫星。 $\mathrm{A} 、 \mathrm{~B} 、 \mathrm{C}$ 三个物体所受地球的万有引力大小分别为 $F A$ 、 $F B$ 和 $F C$; 它们绕地心做圆周运动的线速度大小分别为 $v A 、 v B$ 和 $v C$; 角速度大小分别为 $\omega A 、 \omega B$ 和 $\omega C$; 向心加速度大小分别为 $a A 、 a B$ 和 $a C$ 。则以下分析中正确的是 ( ) A: $F B>F C>F A$ B: $v B>v C>v A$ C: $\omega B>\omega C>\omega A$ D: $a B>a C>a A$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在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_1198

Plotting the position of the Sun in the sky at the same time every day, you get an interesting figure-ofeight shape known as an analemma (see Figure 1). For observers in the Northern hemisphere, you might expect to always see the Sun due South at midday, however on some days the Sun has already passed through that bearing and on others it needs a few more minutes before it gets there. This is due to two effects: the axial tilt of the Earth, and the fact the Earth's orbit is not perfectly circular [figure1] Figure 1: The analemma above was composed from images taken every few days at noon near the village of Callanish in the Outer Hebrides in Scotland. In the foreground are the Callanish Stones and the main photo was taken on the winter solstice (when the maximum angle the Sun reaches above the horizon is the lowest of the year, so is at the bottom of the analemma). Credit: Giuseppe Petricca. The vertical co-ordinate of a point in the analemma is entirely determined by the Earth's axial tilt. This is known as the solar declination, $\delta$, and varies sinusoidally throughout the year. The horizontal coordinate of a point in the analemma is determined by a combination of the Earth's axial tilt and the eccentricity of the Earth's orbit. Both of these individually vary sinusoidally, but the superposition of the two is no longer sinusoidal. We will define $\alpha$ as the angle between due South and the Sun at local midday as seen from Oxford, where a positive value means the Sun has already passed through due South (so is on the right of the figure above) whilst a negative value means the Sun has yet to pass through due South. If $\alpha_{\text {tilt }}$ is the contribution due to the axial tilt and $\alpha_{\text {ecc }}$ is the contribution due to the Earth's orbital eccentricity, then $$ \alpha=\alpha_{\text {tilt }}+\alpha_{\text {ecc }} $$ If the angle of the axial tilt is $\varepsilon$ and the eccentricity of the Earth's orbit is $e$, and we assume that both are small enough that the sinusoidal approximation of $\delta, \alpha_{\text {tilt }}$, and $\alpha_{\text {ecc }}$ apply, then we find the following boundary conditions: - $\delta$ has a period of 1 year, an amplitude of $\varepsilon$, is maximum at the summer solstice (21 $21^{\text {st }}$ June) and minimum at the winter solstice $\left(21^{\text {st }}\right.$ December $)$ - $\alpha_{\text {tilt }}$ has a period of 0.5 years, an amplitude (in radians) of $\tan ^{2}(\varepsilon / 2)$, is zero at the solstices and the equinoxes (vernal equinox $=21^{\text {st }}$ March, autumnal equinox $=21^{\text {st }}$ September), and (using our sign convention) positive just after the vernal equinox - $\alpha_{\text {ecc }}$ has a period of 1 year, an amplitude (in radians) of $2 e$, is zero at the perihelion (4 $4^{\text {th }}$ January) and the aphelion ( $6^{\text {th }}$ July), and (using our sign convention) negative just after the perihelion Given the $n^{\text {th }}$ day of the year, a value can be calculated for $\delta$ and $\alpha$, and these are the co-ordinates for the analemma (it is drawn by these parametric equations). For the Earth, $\varepsilon=23.44^{\circ}$ and $e=0.0167$. Consider an alternative version of Earth, known as Earth 2.0. On this planet, the year is unchanged and the perihelion and aphelion are at the same time, but it has a different axial tilt, a different orbital eccentricity, and a different month for the vernal equinox (although it is still on the $21^{\text {st }}$ day of that month). The analemma as viewed from Earth 2.0 is show in Figure 2 below. [figure2] Figure 2: The analemma of the Sun at midday as seen by an observer on Earth 2.0. In this situation, $\alpha$ ranges from -26 mins 47 secs to 18 mins 56 secs. The circled letters correspond to the same (unknown) day of each month (for example $5^{\text {th }}$ Jan, $5^{\text {th }}$ Feb, $5^{\text {th }}$ March etc.). Credit: Bob Urschel.a. Although $\alpha$ is really an angle in radians (where $2 \pi$ radians $=360^{\circ}$ ), it is normally more useful to convert it into time units (essentially the time since the Sun was due South, or the time until the Sun reaches due South). Taking the mean solar day to be exactly 24 hours: ii. Determine equations for $\delta$ (in degrees) and tilt and ecc (both in minutes) as a function of the day of the year, $n$. Take $n=1$ to be 1 st January and $n=365$ to be 31 st December.

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: Plotting the position of the Sun in the sky at the same time every day, you get an interesting figure-ofeight shape known as an analemma (see Figure 1). For observers in the Northern hemisphere, you might expect to always see the Sun due South at midday, however on some days the Sun has already passed through that bearing and on others it needs a few more minutes before it gets there. This is due to two effects: the axial tilt of the Earth, and the fact the Earth's orbit is not perfectly circular [figure1] Figure 1: The analemma above was composed from images taken every few days at noon near the village of Callanish in the Outer Hebrides in Scotland. In the foreground are the Callanish Stones and the main photo was taken on the winter solstice (when the maximum angle the Sun reaches above the horizon is the lowest of the year, so is at the bottom of the analemma). Credit: Giuseppe Petricca. The vertical co-ordinate of a point in the analemma is entirely determined by the Earth's axial tilt. This is known as the solar declination, $\delta$, and varies sinusoidally throughout the year. The horizontal coordinate of a point in the analemma is determined by a combination of the Earth's axial tilt and the eccentricity of the Earth's orbit. Both of these individually vary sinusoidally, but the superposition of the two is no longer sinusoidal. We will define $\alpha$ as the angle between due South and the Sun at local midday as seen from Oxford, where a positive value means the Sun has already passed through due South (so is on the right of the figure above) whilst a negative value means the Sun has yet to pass through due South. If $\alpha_{\text {tilt }}$ is the contribution due to the axial tilt and $\alpha_{\text {ecc }}$ is the contribution due to the Earth's orbital eccentricity, then $$ \alpha=\alpha_{\text {tilt }}+\alpha_{\text {ecc }} $$ If the angle of the axial tilt is $\varepsilon$ and the eccentricity of the Earth's orbit is $e$, and we assume that both are small enough that the sinusoidal approximation of $\delta, \alpha_{\text {tilt }}$, and $\alpha_{\text {ecc }}$ apply, then we find the following boundary conditions: - $\delta$ has a period of 1 year, an amplitude of $\varepsilon$, is maximum at the summer solstice (21 $21^{\text {st }}$ June) and minimum at the winter solstice $\left(21^{\text {st }}\right.$ December $)$ - $\alpha_{\text {tilt }}$ has a period of 0.5 years, an amplitude (in radians) of $\tan ^{2}(\varepsilon / 2)$, is zero at the solstices and the equinoxes (vernal equinox $=21^{\text {st }}$ March, autumnal equinox $=21^{\text {st }}$ September), and (using our sign convention) positive just after the vernal equinox - $\alpha_{\text {ecc }}$ has a period of 1 year, an amplitude (in radians) of $2 e$, is zero at the perihelion (4 $4^{\text {th }}$ January) and the aphelion ( $6^{\text {th }}$ July), and (using our sign convention) negative just after the perihelion Given the $n^{\text {th }}$ day of the year, a value can be calculated for $\delta$ and $\alpha$, and these are the co-ordinates for the analemma (it is drawn by these parametric equations). For the Earth, $\varepsilon=23.44^{\circ}$ and $e=0.0167$. Consider an alternative version of Earth, known as Earth 2.0. On this planet, the year is unchanged and the perihelion and aphelion are at the same time, but it has a different axial tilt, a different orbital eccentricity, and a different month for the vernal equinox (although it is still on the $21^{\text {st }}$ day of that month). The analemma as viewed from Earth 2.0 is show in Figure 2 below. [figure2] Figure 2: The analemma of the Sun at midday as seen by an observer on Earth 2.0. In this situation, $\alpha$ ranges from -26 mins 47 secs to 18 mins 56 secs. The circled letters correspond to the same (unknown) day of each month (for example $5^{\text {th }}$ Jan, $5^{\text {th }}$ Feb, $5^{\text {th }}$ March etc.). Credit: Bob Urschel. problem: a. Although $\alpha$ is really an angle in radians (where $2 \pi$ radians $=360^{\circ}$ ), it is normally more useful to convert it into time units (essentially the time since the Sun was due South, or the time until the Sun reaches due South). Taking the mean solar day to be exactly 24 hours: ii. Determine equations for $\delta$ (in degrees) and tilt and ecc (both in minutes) as a function of the day of the year, $n$. Take $n=1$ to be 1 st January and $n=365$ to be 31 st December. 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_987

Which of the following cities will experience the longest day during June? A: Edinburgh (longitude $=3.2^{\circ} \mathrm{W}$, latitude $=56.0^{\circ} \mathrm{N}$ ) B: Rome (longitude $=12.5^{\circ} \mathrm{E}$, latitude $=41.9^{\circ} \mathrm{N}$ ) C: Nairobi (longitude $=36.8^{\circ} \mathrm{E}$, latitude $=1.3^{\circ} \mathrm{S}$ ) D: Sydney (longitude $=58.4^{\circ} \mathrm{W}$, latitude $=34.6^{\circ} \mathrm{S}$ )

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: Which of the following cities will experience the longest day during June? A: Edinburgh (longitude $=3.2^{\circ} \mathrm{W}$, latitude $=56.0^{\circ} \mathrm{N}$ ) B: Rome (longitude $=12.5^{\circ} \mathrm{E}$, latitude $=41.9^{\circ} \mathrm{N}$ ) C: Nairobi (longitude $=36.8^{\circ} \mathrm{E}$, latitude $=1.3^{\circ} \mathrm{S}$ ) D: Sydney (longitude $=58.4^{\circ} \mathrm{W}$, latitude $=34.6^{\circ} \mathrm{S}$ ) You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

SC

Astronomy

EN

text-only

Astronomy_1168

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

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 Sun is our closest star so it is arguably the most studied. Results from detailed observations of how sound waves propagate through the plasma of the Sun allow us to get a sense of the general structure of the Sun, with an upper layer of convecting plasma (leading to the 'bubbling' we see with granulation on the surface) and radiative heat transfer below, including a central core region where the pressure and temperature are large enough for nuclear fusion to occur (see Figure 3). [figure1] Figure 3: Left: The general structure of the Sun, with a core in which all the nuclear reactions take place, and convective cells in an outer layer. Credit: Dmitri Pogosyan / University of Alberta. Right: The fraction of the Sun's mass and the contribution to the Sun's luminosity as a function of solar radius as determined from detailed computer simulations. Essentially all of the nuclear reactions creating the photons for the Sun's luminosity take place in a core with a radius of $0.20 R_{\odot}$ and a mass of $0.35 M_{\odot}$. Credit: Kevin France / University of Colorado. Estimating the conditions in the cores of stars is an important aspect of constructing stellar models. This question explores some of the equations governing stellar structure and estimates the central temperature and pressure of the Sun. The primary nuclear fusion pathway responsible for much of the Sun's luminous output is called the proton-proton chain (p-p chain). All of the most common steps are shown in Figure 4. [figure2] Figure 4: An overview of the all the steps in the most common form of the p-p chain, turning four protons into one helium-4 nucleus plus some light particles and photons. Credit: Wikipedia. The net reaction of the p-p chain is $$ 4_{1}^{1} \mathrm{H} \rightarrow{ }_{2}^{4} \mathrm{He}+2 e^{+}+2 \nu_{e}+2 \gamma . $$ The rate-limiting step is the first one $\left({ }_{1}^{1} \mathrm{H}+{ }_{1}^{1} \mathrm{H}\right)$ as the probability of interaction is so low, since it requires the decay of a proton into a neutron and so involves the weak nuclear force. Nuclear physics allow us to calculate the reaction rate coefficient, $R$, energy produced per reaction, $Q$, and hence the energy generation rate, $q$. The reaction rate is related to the number of particles that have enough energy to undergo quantum tunnelling, and the distribution as a function of energy is known as the Gamow peak, with the top of the curve at energy $E_{0}$. For two nuclei, ${ }_{Z_{i}}^{A_{i}} C_{i}$ and ${ }_{Z_{j}}^{A_{j}} C_{j}$, with mass fractions $X_{i}$ and $X_{j}$, then to first order and ignoring electron screening, $$ R=\frac{4}{3^{2.5} \pi^{2}} \frac{h}{\mu_{r} m_{p}} \frac{4 \pi \varepsilon_{0}}{Z_{i} Z_{j} e^{2}} S\left(E_{0}\right) \tau^{2} e^{-\tau}, \quad \text { where } \quad \mu_{r}=\frac{A_{i} A_{j}}{A_{i}+A_{j}} $$ and $S\left(E_{0}\right)$ measures the probability of interaction at the maximum of the Gamow peak whilst $\tau$ is a characteristic width of the Gamow peak, $$ \tau=\frac{3 E_{0}}{k_{B} T} \quad \text { where } \quad E_{0}=\left(\frac{b k_{B} T}{2}\right)^{2 / 3} \quad \text { given } \quad b=\sqrt{\frac{\mu_{r} m_{p}}{2}} \frac{\pi Z_{i} Z_{j} e^{2}}{h \varepsilon_{0}} $$ Here $h$ is Planck's constant, $\varepsilon_{0}$ is the permittivity of free space and $e$ is the elementary charge (the charge on a proton). Finally, the energy generation rate per unit mass is $$ q=\frac{\rho}{m_{p}^{2}}\left(\frac{1}{1+\delta_{i j}}\right) \frac{X_{i} X_{j}}{A_{i} A_{j}} R Q $$ where $\delta_{i j}$ is the Kronecker delta, so equals 1 when $i=j$ and 0 otherwise. Evaluating the fundamental constants and defining $T_{6} \equiv \frac{T}{10^{6} \mathrm{~K}}$ gives $$ \tau=42.59\left[Z_{i}^{2} Z_{j}^{2} \mu_{r} T_{6}^{-1}\right]^{1 / 3}, $$ whilst for the proton-proton interaction $Q=13.366 \mathrm{MeV}$ (half the overall energy of the p-p chain), $Z_{i}=Z_{j}=A_{i}=A_{j}=\delta_{i j}=1$, and $S\left(E_{0}\right)$ is $4.01 \times 10^{-50} \mathrm{keV} \mathrm{m}^{2}$ (Adelberger et al. 2011), so $R=6.55 \times 10^{-43} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \mathrm{~m}^{3} \mathrm{~s}^{-1}$ and $q=0.251 \rho X^{2} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \mathrm{~W} \mathrm{~kg}^{-1}$. problem: a. Let $r$ denote distance from the centre of a star. We define the variables $\rho(r), p(r)$ and $T(r)$ to be the density, pressure and temperature at radius $r$ respectively, and $m(r)$ to be the mass enclosed within radius $r$. We will now try and derive an estimate for the pressure at the centre of the Sun. i. By considering forces on a box of height $d r$ at radius $r$, show that $d p / d r=-\rho G m / r^{2}$. 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_172

如图所示, $\mathrm{A}$ 和 $\mathrm{B}$ 两行星绕同一恒星 $\mathrm{C}$ 做圆周运动, 旋转方向相同, $\mathrm{A}$ 行星的周期为 $T_{1}, \mathrm{~B}$ 行星的周期为 $T_{2}$, 某一时刻两行星相距最近, 则 $(\quad)$ [图1] A: 经过 $T_{1}+T_{2}$ 两行星再次相距最近 B: 经过 $\frac{T_{1} T_{2}}{T_{2}-T_{1}}$ 两行星再次相距最近 C: 经过 $\frac{T_{1}+T_{2}}{2}$ 两行星相距最远 D: 经过 $\frac{T_{1} T_{2}}{T_{2}-T_{1}}$ 两行星相距最远

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 如图所示, $\mathrm{A}$ 和 $\mathrm{B}$ 两行星绕同一恒星 $\mathrm{C}$ 做圆周运动, 旋转方向相同, $\mathrm{A}$ 行星的周期为 $T_{1}, \mathrm{~B}$ 行星的周期为 $T_{2}$, 某一时刻两行星相距最近, 则 $(\quad)$ [图1] A: 经过 $T_{1}+T_{2}$ 两行星再次相距最近 B: 经过 $\frac{T_{1} T_{2}}{T_{2}-T_{1}}$ 两行星再次相距最近 C: 经过 $\frac{T_{1}+T_{2}}{2}$ 两行星相距最远 D: 经过 $\frac{T_{1} T_{2}}{T_{2}-T_{1}}$ 两行星相距最远 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy_345

2021 年 5 月 15 日, “天问一号”着陆器成功着陆于火星乌托邦平原南部预选着陆区,我国首次火星探测任务着陆火星取得圆满成功。如图为“天问一号”的地火转移轨道，为了节省燃料, 我们在火星与地球之间相对合适位置时发射“天问一号”。将火星与地球绕太阳的运动简化为在同一平面、沿同一方向的匀速圆周运动。下列说法正确的是 ( ) [图1] A: 火星的公转周期大于地球的公转周期 B: 火星公转的向心加速度大于地球公转的向心加速度 C: “天问一号”在地火转移轨道上运动的周期小于地球绕太阳运动的周期 D: “天问一号”从 $A$ 点运动到 $C$ 点的过程中处于加速状态

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 5 月 15 日, “天问一号”着陆器成功着陆于火星乌托邦平原南部预选着陆区,我国首次火星探测任务着陆火星取得圆满成功。如图为“天问一号”的地火转移轨道，为了节省燃料, 我们在火星与地球之间相对合适位置时发射“天问一号”。将火星与地球绕太阳的运动简化为在同一平面、沿同一方向的匀速圆周运动。下列说法正确的是 ( ) [图1] A: 火星的公转周期大于地球的公转周期 B: 火星公转的向心加速度大于地球公转的向心加速度 C: “天问一号”在地火转移轨道上运动的周期小于地球绕太阳运动的周期 D: “天问一号”从 $A$ 点运动到 $C$ 点的过程中处于加速状态 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在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_612

如图所示, 飞船在地面指挥控制中心的控制下, 由近地点圆形轨道 $A$, 经粗圆轨道 $B$ 转变到远地点的圆轨道 $C$. 轨道 $A$ 与轨道 $B$ 相切于 $P$ 点, 轨道 $B$ 与轨道 $C$ 相切于 $Q$ 点,以下说法正确的是 ( ) [图1] A: 卫星在轨道 $B$ 上由 $P$ 向 $Q$ 运动的过程中速率越来越小 B: 卫星在轨道 $C$ 上经过 $Q$ 点的速率大于在轨道 $A$ 上经过 $P$ 点的速率 C: 卫星在轨道 $B$ 上经过 $P$ 点的加速度与在轨道 $A$ 上经过 $P$ 点的加速度是相等的 D: 卫星在轨道 $B$ 上经过 $Q$ 点的速度小于卫星在轨道 $\mathrm{C}$ 上经过 $\mathrm{Q}$ 点速度

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 如图所示, 飞船在地面指挥控制中心的控制下, 由近地点圆形轨道 $A$, 经粗圆轨道 $B$ 转变到远地点的圆轨道 $C$. 轨道 $A$ 与轨道 $B$ 相切于 $P$ 点, 轨道 $B$ 与轨道 $C$ 相切于 $Q$ 点,以下说法正确的是 ( ) [图1] A: 卫星在轨道 $B$ 上由 $P$ 向 $Q$ 运动的过程中速率越来越小 B: 卫星在轨道 $C$ 上经过 $Q$ 点的速率大于在轨道 $A$ 上经过 $P$ 点的速率 C: 卫星在轨道 $B$ 上经过 $P$ 点的加速度与在轨道 $A$ 上经过 $P$ 点的加速度是相等的 D: 卫星在轨道 $B$ 上经过 $Q$ 点的速度小于卫星在轨道 $\mathrm{C}$ 上经过 $\mathrm{Q}$ 点速度 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

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Astronomy

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multi-modal

Astronomy_1003

An asteroid in a circular orbit around the Sun is at its closest to Earth every 300 days. What is its orbital speed? Assume the Earth's orbit is also circular and that both orbit in the same direction. A: $21 \mathrm{~km} \mathrm{~s}^{-1}$ B: $28 \mathrm{~km} \mathrm{~s}^{-1}$ C: $32 \mathrm{~km} \mathrm{~s}^{-1}$ D: $39 \mathrm{~km} \mathrm{~s}^{-1}$

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: An asteroid in a circular orbit around the Sun is at its closest to Earth every 300 days. What is its orbital speed? Assume the Earth's orbit is also circular and that both orbit in the same direction. A: $21 \mathrm{~km} \mathrm{~s}^{-1}$ B: $28 \mathrm{~km} \mathrm{~s}^{-1}$ C: $32 \mathrm{~km} \mathrm{~s}^{-1}$ D: $39 \mathrm{~km} \mathrm{~s}^{-1}$ You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

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Astronomy_922

In Autumn 2022, NASA plans to launch a practice mission in preparation for sending humans back to the Moon. What is the name of the programme this is part of? A: Artemis B: Athena C: Diana D: Orion

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 Autumn 2022, NASA plans to launch a practice mission in preparation for sending humans back to the Moon. What is the name of the programme this is part of? A: Artemis B: Athena C: Diana D: Orion 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_264

一球状行星的自转与地球自转的运动情况相似, 此行星的一昼夜为 $a$ 秒, 在星球上的不同位置用弹簧科测量同一物体的重力, 在此星球赤道上称得的重力是在北极处的 $b$倍 ( $b$ 小于 1$)$, 万有引力常量为 $\mathrm{G}$, 则此行星的平均密度为 $(\quad)$ A: $\frac{3 \pi}{G a^{2}(1-b)}$ B: $\frac{3 \pi}{G a^{2} b}$ C: $\frac{30 \pi}{G a^{2}(1-b)}$ D: $\frac{30 \pi}{G a^{2} b}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 一球状行星的自转与地球自转的运动情况相似, 此行星的一昼夜为 $a$ 秒, 在星球上的不同位置用弹簧科测量同一物体的重力, 在此星球赤道上称得的重力是在北极处的 $b$倍 ( $b$ 小于 1$)$, 万有引力常量为 $\mathrm{G}$, 则此行星的平均密度为 $(\quad)$ A: $\frac{3 \pi}{G a^{2}(1-b)}$ B: $\frac{3 \pi}{G a^{2} b}$ C: $\frac{30 \pi}{G a^{2}(1-b)}$ D: $\frac{30 \pi}{G a^{2} b}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

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Astronomy_387

已知某星球的近地卫星和同步卫星的周期分别为 $T$ 和 $8 T$, 星球半径为 $R$, 引力常量为 $G$, 星球赤道上有一静止的质量为 $m$ 的物体, 若把星球视为一个质量均匀的球体,则下列说法不正确的是（） A: 该星球的质量为 $\frac{4 \pi^{2} R^{3}}{G T^{2}}$ B: 该星球的密度为 $\frac{3 \pi}{G T^{2}}$ C: 该星球同步卫星的轨道半径为 $4 R$ D: 赤道对该物体的支持力大小为 $\frac{63 \pi^{2} m R}{64 T^{2}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 已知某星球的近地卫星和同步卫星的周期分别为 $T$ 和 $8 T$, 星球半径为 $R$, 引力常量为 $G$, 星球赤道上有一静止的质量为 $m$ 的物体, 若把星球视为一个质量均匀的球体,则下列说法不正确的是（） A: 该星球的质量为 $\frac{4 \pi^{2} R^{3}}{G T^{2}}$ B: 该星球的密度为 $\frac{3 \pi}{G T^{2}}$ C: 该星球同步卫星的轨道半径为 $4 R$ D: 赤道对该物体的支持力大小为 $\frac{63 \pi^{2} m R}{64 T^{2}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy_394

某同学设计了一个滑梯游戏装置, 如图所示, 一光滑轨道 $A O$ 固定在水平桌面上, $O$ 点在桌面右侧边缘上。以 $O$ 点为圆心的 $\frac{1}{4}$ 光滑圆弧轨道 $B D$ 坚直固定在桌子的右侧, $C$ 点为圆弧轨道 $B D$ 的中点。若宇航员利用该游戏装置分别在地球表面和火星表面进行模拟实验, 将小球放在光滑轨道 $A O$ 上某点由静止下滑, 小球越过 $O$ 点后飞出, 落在光滑圆弧轨道 $B D$ 上。忽略空气阻力, 已知地球表面的重力加速度大小为 $\mathrm{g}$, 火星的质量约为地球质量的 $\frac{1}{9}$, 火星的半径约为地球半径的 $\frac{1}{2}$ 。在地球表面或在火星表面上, 下列说法正确的是（ ） [图1] A: 若小球恰能打到 $C$ 点, 则击中 $C$ 点时的速度方向与圆弧面垂直 B: 小球释放点越低, 小球落到圆弧上时动能就越小 C: 根据题目的条件可以得出火星表面的重力加速度大小 $g_{\text {火 }}=\frac{9}{4} g$ D: 在地球和火星进行模拟实验时, 若都从光滑轨道上同一位置释放小球, 则小球将落在圆弧上的同一点

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 某同学设计了一个滑梯游戏装置, 如图所示, 一光滑轨道 $A O$ 固定在水平桌面上, $O$ 点在桌面右侧边缘上。以 $O$ 点为圆心的 $\frac{1}{4}$ 光滑圆弧轨道 $B D$ 坚直固定在桌子的右侧, $C$ 点为圆弧轨道 $B D$ 的中点。若宇航员利用该游戏装置分别在地球表面和火星表面进行模拟实验, 将小球放在光滑轨道 $A O$ 上某点由静止下滑, 小球越过 $O$ 点后飞出, 落在光滑圆弧轨道 $B D$ 上。忽略空气阻力, 已知地球表面的重力加速度大小为 $\mathrm{g}$, 火星的质量约为地球质量的 $\frac{1}{9}$, 火星的半径约为地球半径的 $\frac{1}{2}$ 。在地球表面或在火星表面上, 下列说法正确的是（ ） [图1] A: 若小球恰能打到 $C$ 点, 则击中 $C$ 点时的速度方向与圆弧面垂直 B: 小球释放点越低, 小球落到圆弧上时动能就越小 C: 根据题目的条件可以得出火星表面的重力加速度大小 $g_{\text {火 }}=\frac{9}{4} g$ D: 在地球和火星进行模拟实验时, 若都从光滑轨道上同一位置释放小球, 则小球将落在圆弧上的同一点 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy_299

2013 年 5 月 2 日凌晨 0 时 06 分, 我国“中星 11 号”通信卫星发射成功, “中星 11 号”是一颗地球同步卫星, 它主要用于为亚太地区等区域用户提供商业通信服务, 图 2 为发射过程的示意图, 先将卫星发射至近地圆轨道 1 , 速度为 $v_{1}$, 然后经点火, 使其沿粗圆轨道 2 运行, 在椭圆轨道上 $P 、 Q$ 点的速度分别为 $v_{2 \mathrm{P}}$ 和 $v_{2 \mathrm{Q}}$, 最后再一次点火, 将卫星送入同步圆轨道 3 , 速度为 $v_{3}$, 轨道 $1 、 2$ 相切于 $Q$ 点, 轨道 $2 、 3$ 相切于 $P$ 点, 则当卫星分别在 1、2、3 轨道上正常运行时, 以下说法正确的是（ ） [图1] A: 四个速率的大小顺序为: $v_{2 \mathrm{Q}}>v_{2 \mathrm{P}}<v_{3}<v_{1}, v_{3}$ 与 $v_{2 \mathrm{Q}}$ 的大小不能比较 B: 卫星在轨道 3 上的角速度大于在轨道 1 上的角速度 C: 卫星在轨道 1 上经过 $Q$ 点时的加速度小于它在轨道 2 上经过 $Q$ 点时的加速度 D: 卫星在轨道 1 上的机械能小于它在轨道 3 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。 这是一个单选题（只有一个正确答案）。 问题: 2013 年 5 月 2 日凌晨 0 时 06 分, 我国“中星 11 号”通信卫星发射成功, “中星 11 号”是一颗地球同步卫星, 它主要用于为亚太地区等区域用户提供商业通信服务, 图 2 为发射过程的示意图, 先将卫星发射至近地圆轨道 1 , 速度为 $v_{1}$, 然后经点火, 使其沿粗圆轨道 2 运行, 在椭圆轨道上 $P 、 Q$ 点的速度分别为 $v_{2 \mathrm{P}}$ 和 $v_{2 \mathrm{Q}}$, 最后再一次点火, 将卫星送入同步圆轨道 3 , 速度为 $v_{3}$, 轨道 $1 、 2$ 相切于 $Q$ 点, 轨道 $2 、 3$ 相切于 $P$ 点, 则当卫星分别在 1、2、3 轨道上正常运行时, 以下说法正确的是（ ） [图1] A: 四个速率的大小顺序为: $v_{2 \mathrm{Q}}>v_{2 \mathrm{P}}<v_{3}<v_{1}, v_{3}$ 与 $v_{2 \mathrm{Q}}$ 的大小不能比较 B: 卫星在轨道 3 上的角速度大于在轨道 1 上的角速度 C: 卫星在轨道 1 上经过 $Q$ 点时的加速度小于它在轨道 2 上经过 $Q$ 点时的加速度 D: 卫星在轨道 1 上的机械能小于它在轨道 3 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应为以下选项之一：[A, B, C, D]

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Astronomy_609

如图, $O$ 为地球的球心, 可视为质点的卫星 $\mathrm{A} 、 \mathrm{~B}$ 在同一平面内绕地球做同向匀速圆周运动。某时刻, 连线 $B A$ 垂直于 $B O, B A$ 与 $O A$ 的夹角 $\theta=30^{\circ}$, 卫星 $\mathrm{B}$ 的公转周期为 2 小时, 则卫星 $\mathrm{A}$ 的公转周期为 [图1] A: 8 小时 B: 5.6 小时 C: 4 小时 D: 3.4 小时

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, $O$ 为地球的球心, 可视为质点的卫星 $\mathrm{A} 、 \mathrm{~B}$ 在同一平面内绕地球做同向匀速圆周运动。某时刻, 连线 $B A$ 垂直于 $B O, B A$ 与 $O A$ 的夹角 $\theta=30^{\circ}$, 卫星 $\mathrm{B}$ 的公转周期为 2 小时, 则卫星 $\mathrm{A}$ 的公转周期为 [图1] A: 8 小时 B: 5.6 小时 C: 4 小时 D: 3.4 小时 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在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_566

利用水流和太阳能发电, 可以为人类提供清洁能源。已知太阳光垂直照射到地面上时的辐射功率 $P_{0}=1.0 \times 10^{3} \mathrm{~W} / \mathrm{m}^{2}$, 地球表面的重力加速度取 $g=10 \mathrm{~m} / \mathrm{s}^{2}$, 水的密度 $\rho=1.0 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ 。 三峡水电站发电机输出的电压为 $18 \mathrm{kV}$ 。若采用 $500 \mathrm{kV}$ 直流电向某地区输电 $5.0 \times 10^{6} \mathrm{~kW}$, 要求输电线上损耗的功率不高于输送功率的 $5 \%$, 求输电线总电阻的最大值;

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个数值。 问题: 利用水流和太阳能发电, 可以为人类提供清洁能源。已知太阳光垂直照射到地面上时的辐射功率 $P_{0}=1.0 \times 10^{3} \mathrm{~W} / \mathrm{m}^{2}$, 地球表面的重力加速度取 $g=10 \mathrm{~m} / \mathrm{s}^{2}$, 水的密度 $\rho=1.0 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ 。 三峡水电站发电机输出的电压为 $18 \mathrm{kV}$ 。若采用 $500 \mathrm{kV}$ 直流电向某地区输电 $5.0 \times 10^{6} \mathrm{~kW}$, 要求输电线上损耗的功率不高于输送功率的 $5 \%$, 求输电线总电阻的最大值; 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 请记住，你的答案应以$ \Omega$为单位计算，但在给出最终答案时，请不要包含单位。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。

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Astronomy_556

2022 年 10 月 31 日, “梦天实验舱”发射任务取得圆满成功！中国空间空间站将形成三舱“T”字型基本构型。假定空间站在距地面 $450 \mathrm{~km}$ 高度处做理想的匀速圆周运动, 某时刻“北斗” 系统中的中轨道卫星 A 与空间站相距最近如图所示, 该中轨道卫星 A 距地面高度为 $2.1 \times 10^{7} \mathrm{~m}$, 地球半径为 $6.4 \times 10^{6} \mathrm{~m}$, 卫星 $\mathrm{A}$ 和空间站的运行轨道在同一平面内且运行方向相同, 则从图示位置往后开始计数 (不包括图示位置), 在卫星 A 运行一周时间内，空间站与 A 相距最近的次数为（） [图1] A: 7 次 B: 8 次 C: 9 次 D: 14 次

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 2022 年 10 月 31 日, “梦天实验舱”发射任务取得圆满成功！中国空间空间站将形成三舱“T”字型基本构型。假定空间站在距地面 $450 \mathrm{~km}$ 高度处做理想的匀速圆周运动, 某时刻“北斗” 系统中的中轨道卫星 A 与空间站相距最近如图所示, 该中轨道卫星 A 距地面高度为 $2.1 \times 10^{7} \mathrm{~m}$, 地球半径为 $6.4 \times 10^{6} \mathrm{~m}$, 卫星 $\mathrm{A}$ 和空间站的运行轨道在同一平面内且运行方向相同, 则从图示位置往后开始计数 (不包括图示位置), 在卫星 A 运行一周时间内，空间站与 A 相距最近的次数为（） [图1] A: 7 次 B: 8 次 C: 9 次 D: 14 次 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy_514

2023 年 10 月 26 日消息, 韦伯望远镜首次检测到恒星合并后啼（tellurium）等重元素的存在, 可以帮助天文学家探究地球生命起源的奥秘。韦伯望远镜位于“拉格朗日 $L_{2}$点”上, 跟随地球一起围绕太阳做圆周运动, 图中的虚线圆周表示地球和韦伯望远镜绕太阳运动的轨道, 韦伯望远镜和地球相对位置总是保持不变。已知太阳质量为 $M_{1}$ 、地球质量为 $M_{2}$, 地球到太阳的距离为 $R$, 用 $l$ 表示韦伯望远镜到地球的距离，把太阳、地球都看做是质点。由于 $\frac{l}{R}$ 的值很小, 根据数学知识可以解出 $l \approx \sqrt{\frac{M_{2}}{3 M_{1}}} R$, 你可能不知道这个解是用怎样的数学方法求出的, 但根据物理知识你可以得出这个解对应的方程式为 $(\quad)$ [图1] A: $\frac{R+l}{R^{3}}-\frac{1}{(R+l)^{2}}=\frac{1}{l^{2}} \cdot \frac{M_{2}}{M_{1}}$ B: $\frac{R+l}{R^{3}}+\frac{1}{(R+l)^{2}}=\frac{1}{l^{2}} \cdot \frac{M_{2}}{M_{1}}$ C: $\frac{R+l}{R^{3}}-\frac{1}{(R+l)^{2}}=\frac{1}{l^{2}} \cdot \frac{M_{2}}{3 M_{1}}$ D: $\frac{R+l}{R^{3}}+\frac{1}{(R+l)^{2}}=\frac{1}{l^{2}} \cdot \frac{M_{2}}{3 M_{1}}$

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: 2023 年 10 月 26 日消息, 韦伯望远镜首次检测到恒星合并后啼（tellurium）等重元素的存在, 可以帮助天文学家探究地球生命起源的奥秘。韦伯望远镜位于“拉格朗日 $L_{2}$点”上, 跟随地球一起围绕太阳做圆周运动, 图中的虚线圆周表示地球和韦伯望远镜绕太阳运动的轨道, 韦伯望远镜和地球相对位置总是保持不变。已知太阳质量为 $M_{1}$ 、地球质量为 $M_{2}$, 地球到太阳的距离为 $R$, 用 $l$ 表示韦伯望远镜到地球的距离，把太阳、地球都看做是质点。由于 $\frac{l}{R}$ 的值很小, 根据数学知识可以解出 $l \approx \sqrt{\frac{M_{2}}{3 M_{1}}} R$, 你可能不知道这个解是用怎样的数学方法求出的, 但根据物理知识你可以得出这个解对应的方程式为 $(\quad)$ [图1] A: $\frac{R+l}{R^{3}}-\frac{1}{(R+l)^{2}}=\frac{1}{l^{2}} \cdot \frac{M_{2}}{M_{1}}$ B: $\frac{R+l}{R^{3}}+\frac{1}{(R+l)^{2}}=\frac{1}{l^{2}} \cdot \frac{M_{2}}{M_{1}}$ C: $\frac{R+l}{R^{3}}-\frac{1}{(R+l)^{2}}=\frac{1}{l^{2}} \cdot \frac{M_{2}}{3 M_{1}}$ D: $\frac{R+l}{R^{3}}+\frac{1}{(R+l)^{2}}=\frac{1}{l^{2}} \cdot \frac{M_{2}}{3 M_{1}}$ 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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multi-modal

Astronomy_256

密度均匀的球体半径为 $R$ 、质量为 $m$, 现从球体 $\mathrm{A}$ 中挖去直径为 $R$ 的球体 $\mathrm{B}$, 将球体 $\mathrm{B}$ 放置在距离球体 $\mathrm{A}$ 的球心 $O$ 为 $2 R$ 处, 如图所示, 白色部分为挖去后的空心。已知半径为 $R$ 的球体的体积为 $\frac{4}{3} \pi R^{3}$, 引力常量为 $G$, 则球体 $\mathrm{A}$ 剩余部分对球体 $\mathrm{B}$ 的万有引力大小为 ( ) [图1] A: $G \frac{7 m^{2}}{256 R^{2}}$ B: $G \frac{7 m^{2}}{400 R^{2}}$ C: $G \frac{23 m^{2}}{256 R^{2}}$ D: $G \frac{23 m^{2}}{800 R^{2}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 密度均匀的球体半径为 $R$ 、质量为 $m$, 现从球体 $\mathrm{A}$ 中挖去直径为 $R$ 的球体 $\mathrm{B}$, 将球体 $\mathrm{B}$ 放置在距离球体 $\mathrm{A}$ 的球心 $O$ 为 $2 R$ 处, 如图所示, 白色部分为挖去后的空心。已知半径为 $R$ 的球体的体积为 $\frac{4}{3} \pi R^{3}$, 引力常量为 $G$, 则球体 $\mathrm{A}$ 剩余部分对球体 $\mathrm{B}$ 的万有引力大小为 ( ) [图1] A: $G \frac{7 m^{2}}{256 R^{2}}$ B: $G \frac{7 m^{2}}{400 R^{2}}$ C: $G \frac{23 m^{2}}{256 R^{2}}$ D: $G \frac{23 m^{2}}{800 R^{2}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy_1100

Hanny's Voorwerp (Dutch for 'object') is a rare type of astronomical object discovered in 2007 by the school teacher Hanny van Arkel whilst participating as a volunteer in the Galaxy Zoo project. When inspecting the image of the galaxy IC 2497 in the constellation Leo Minor, she observed a bright green blob close to the galaxy. [figure1] Figure 5: HST image of galaxy IC 2497 and the glowing Voorwerp below it. Credit: Keel et al. (2012) \& Galaxy Zoo. Subsequent observations have shown that the galaxy IC 2497 is at a redshift of $z=0.05$, with the Voorwerp at a similar distance and with a projected angular separation of 20 arcseconds from the centre of the galaxy $\left(3600\right.$ arcseconds $\left.=1^{\circ}\right)$. Radio observations suggest that the Voorwerp is a massive cloud of gas, made of ionized hydrogen, with a size of $10 \mathrm{kpc}$ and a mass of $10^{11} \mathrm{M}_{\odot}$. It is probably a cloud of gas that was stripped from the galaxy during a merger with another nearby galaxy. In this question you will explore the cause of the 'glow' of the Voorwerp and will learn about a new type of an astronomical object; a quasar.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: Hanny's Voorwerp (Dutch for 'object') is a rare type of astronomical object discovered in 2007 by the school teacher Hanny van Arkel whilst participating as a volunteer in the Galaxy Zoo project. When inspecting the image of the galaxy IC 2497 in the constellation Leo Minor, she observed a bright green blob close to the galaxy. [figure1] Figure 5: HST image of galaxy IC 2497 and the glowing Voorwerp below it. Credit: Keel et al. (2012) \& Galaxy Zoo. Subsequent observations have shown that the galaxy IC 2497 is at a redshift of $z=0.05$, with the Voorwerp at a similar distance and with a projected angular separation of 20 arcseconds from the centre of the galaxy $\left(3600\right.$ arcseconds $\left.=1^{\circ}\right)$. Radio observations suggest that the Voorwerp is a massive cloud of gas, made of ionized hydrogen, with a size of $10 \mathrm{kpc}$ and a mass of $10^{11} \mathrm{M}_{\odot}$. It is probably a cloud of gas that was stripped from the galaxy during a merger with another nearby galaxy. In this question you will explore the cause of the 'glow' of the Voorwerp and will learn about a new type of an astronomical object; a quasar. 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 \mathrm{~W}, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

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multi-modal

Astronomy_456

如图 (i) 所示, 真空中两正点电荷 $\mathrm{A} 、 \mathrm{~B}$ 固定在 $x$ 轴上, 其中 $\mathrm{A}$ 位于坐标原点。 一质量为 $m$ 、电量为 $q$ (电量远小于 $\mathrm{A} 、 \mathrm{~B}$ ) 的带正电小球 $\mathrm{a}$ 仅在电场力作用下, 以大小为 $v_{0}$ 的初速度从 $x=x_{1}$ 处沿 $x$ 轴正方向运动。取无穷远处势能为零, $\mathrm{a}$ 在 $\mathrm{A} 、 \mathrm{~B}$ 间由于受 $\mathrm{A} 、 \mathrm{~B}$ 的电场力作用而具有的电势能 $E_{p}$ 随位置 $x$ 变化关系如图 (ii) 所示, 图中 $E_{N_{1}}$, $E_{2}$ 均为已知, 且 $\mathrm{a}$ 在 $x=x_{2}$ 处受到的电场力为零。 求 $\mathrm{A} 、 \mathrm{~B}$ 两电荷电场在 $x=x_{1}$ 与 $x=x_{2}$ 两点间的电势差 $U_{12}$; [图1] 图(i) [图2] 图(ii) [图3] 图(iii) [图4] 图(iv)

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 如图 (i) 所示, 真空中两正点电荷 $\mathrm{A} 、 \mathrm{~B}$ 固定在 $x$ 轴上, 其中 $\mathrm{A}$ 位于坐标原点。 一质量为 $m$ 、电量为 $q$ (电量远小于 $\mathrm{A} 、 \mathrm{~B}$ ) 的带正电小球 $\mathrm{a}$ 仅在电场力作用下, 以大小为 $v_{0}$ 的初速度从 $x=x_{1}$ 处沿 $x$ 轴正方向运动。取无穷远处势能为零, $\mathrm{a}$ 在 $\mathrm{A} 、 \mathrm{~B}$ 间由于受 $\mathrm{A} 、 \mathrm{~B}$ 的电场力作用而具有的电势能 $E_{p}$ 随位置 $x$ 变化关系如图 (ii) 所示, 图中 $E_{N_{1}}$, $E_{2}$ 均为已知, 且 $\mathrm{a}$ 在 $x=x_{2}$ 处受到的电场力为零。 求 $\mathrm{A} 、 \mathrm{~B}$ 两电荷电场在 $x=x_{1}$ 与 $x=x_{2}$ 两点间的电势差 $U_{12}$; [图1] 图(i) [图2] 图(ii) [图3] 图(iii) [图4] 图(iv) 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

EX

Astronomy

ZH

multi-modal

Astronomy_776

When a neutron star rotates, it becomes a ... A: Neutron dwarf B: Rotar C: Quasar D: Pulsar

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: When a neutron star rotates, it becomes a ... A: Neutron dwarf B: Rotar C: Quasar D: Pulsar 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_1149

GW170817 was the first gravitational wave event arising from a binary neutron star merger to have been detected by the LIGO \& Virgo experiments, and careful localization of the source meant that the electromagnetic counterpart was quickly found in galaxy NGC 4993 (see Figure 2). Such a combination of two completely separate branches of astronomical observation begins a new era of 'multi-messenger astronomy'. Since the gravitational waves allow an independent measurement of the distance to the host galaxy and the light allows an independent measurement of the recessional speed, this observation allows us to determine a new, independent value of the Hubble constant $H_{0}$. Another way of measuring distances to galaxies is to use the Fundamental Plane (FP) relation, which relies on the assumption there is a fairly tight relation between radius, surface brightness, and velocity dispersion for bulge-dominated galaxies, and is widely used for galaxies like NGC 4993. It can be described by the relation $$ \log \left(\frac{D}{(1+z)^{2}}\right)=-\log R_{e}+\alpha \log \sigma-\beta \log \left\langle I_{r}\right\rangle_{e}+\gamma $$ where $D$ is the distance in Mpc, $R_{e}$ is the effective radius measured in arcseconds, $\sigma$ is the velocity dispersion in $\mathrm{km} \mathrm{s}^{-1},\left\langle I_{r}\right\rangle_{e}$ is the mean intensity inside the effective radius measured in $L_{\odot} \mathrm{pc}^{-2}$, and $\gamma$ is the distance-dependent zero point of the relation. Calibrating the zero point to the Leo I galaxy group, the constants in the FP relation become $\alpha=1.24, \beta=0.82$, and $\gamma=2.194$. Figure 2: Left: The GW170817 signal as measured by the LIGO and Virgo gravitational wave detectors, taken from Abbott $e t$ al. (2017). The normalized amplitude (or strain) is in units of $10^{-21}$. The signal is not visible in the Virgo data due to the direction of the source with respect to the detector's antenna pattern. Right: The optical counterpart of GW170817 in host galaxy NGC 4993, taken from Hjorth et al. (2017). By measuring the amplitude (called strain, $h$ ) and the frequency of the gravitational waves, $f_{\mathrm{GW}}$, one can determine the distance to the source without having to rely on 'standard candles' like Cepheid variables or Type Ia supernovae. For two masses, $m_{1}$ and $m_{2}$, orbiting the centre of mass with separation $a$ with orbital angular velocity $\omega$, then the dimensionless strain parameter $h$ is $$ h \simeq \frac{G}{c^{4}} \frac{1}{r} \mu a^{2} \omega^{2} $$ where $r$ is the luminosity distance, $c$ is the speed of light, $\mu=m_{1} m_{2} / M_{\text {tot }}$ is the reduced mass and $M_{\text {tot }}=m_{1}+m_{2}$ is the total mass. The rate of change of frequency of the gravitational waves (called the 'chirp') from a merging binary can be written as $$ \dot{f}_{\mathrm{GW}}=\frac{96}{5} \pi^{8 / 3}\left(\frac{G \mathcal{M}}{c^{3}}\right)^{5 / 3} f_{\mathrm{GW}}^{11 / 3} $$f. Using the redshift information from NGC 4993 and the gravitational wave distance you have just calculated, determine the Hubble constant $\mathrm{H}_{0}$ in units of $\mathrm{km} \mathrm{s}^{-1} \mathrm{Mpc}^{-1}$, along with its absolute uncertainty. Is this value consistent with the one derived by the HST using Cepheid variables (given in 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: GW170817 was the first gravitational wave event arising from a binary neutron star merger to have been detected by the LIGO \& Virgo experiments, and careful localization of the source meant that the electromagnetic counterpart was quickly found in galaxy NGC 4993 (see Figure 2). Such a combination of two completely separate branches of astronomical observation begins a new era of 'multi-messenger astronomy'. Since the gravitational waves allow an independent measurement of the distance to the host galaxy and the light allows an independent measurement of the recessional speed, this observation allows us to determine a new, independent value of the Hubble constant $H_{0}$. Another way of measuring distances to galaxies is to use the Fundamental Plane (FP) relation, which relies on the assumption there is a fairly tight relation between radius, surface brightness, and velocity dispersion for bulge-dominated galaxies, and is widely used for galaxies like NGC 4993. It can be described by the relation $$ \log \left(\frac{D}{(1+z)^{2}}\right)=-\log R_{e}+\alpha \log \sigma-\beta \log \left\langle I_{r}\right\rangle_{e}+\gamma $$ where $D$ is the distance in Mpc, $R_{e}$ is the effective radius measured in arcseconds, $\sigma$ is the velocity dispersion in $\mathrm{km} \mathrm{s}^{-1},\left\langle I_{r}\right\rangle_{e}$ is the mean intensity inside the effective radius measured in $L_{\odot} \mathrm{pc}^{-2}$, and $\gamma$ is the distance-dependent zero point of the relation. Calibrating the zero point to the Leo I galaxy group, the constants in the FP relation become $\alpha=1.24, \beta=0.82$, and $\gamma=2.194$. Figure 2: Left: The GW170817 signal as measured by the LIGO and Virgo gravitational wave detectors, taken from Abbott $e t$ al. (2017). The normalized amplitude (or strain) is in units of $10^{-21}$. The signal is not visible in the Virgo data due to the direction of the source with respect to the detector's antenna pattern. Right: The optical counterpart of GW170817 in host galaxy NGC 4993, taken from Hjorth et al. (2017). By measuring the amplitude (called strain, $h$ ) and the frequency of the gravitational waves, $f_{\mathrm{GW}}$, one can determine the distance to the source without having to rely on 'standard candles' like Cepheid variables or Type Ia supernovae. For two masses, $m_{1}$ and $m_{2}$, orbiting the centre of mass with separation $a$ with orbital angular velocity $\omega$, then the dimensionless strain parameter $h$ is $$ h \simeq \frac{G}{c^{4}} \frac{1}{r} \mu a^{2} \omega^{2} $$ where $r$ is the luminosity distance, $c$ is the speed of light, $\mu=m_{1} m_{2} / M_{\text {tot }}$ is the reduced mass and $M_{\text {tot }}=m_{1}+m_{2}$ is the total mass. The rate of change of frequency of the gravitational waves (called the 'chirp') from a merging binary can be written as $$ \dot{f}_{\mathrm{GW}}=\frac{96}{5} \pi^{8 / 3}\left(\frac{G \mathcal{M}}{c^{3}}\right)^{5 / 3} f_{\mathrm{GW}}^{11 / 3} $$ problem: f. Using the redshift information from NGC 4993 and the gravitational wave distance you have just calculated, determine the Hubble constant $\mathrm{H}_{0}$ in units of $\mathrm{km} \mathrm{s}^{-1} \mathrm{Mpc}^{-1}$, along with its absolute uncertainty. Is this value consistent with the one derived by the HST using Cepheid variables (given in 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 \mathrm{~km} \mathrm{~s}^{-1} \mathrm{Mpc}^{-1}, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

text-only

Astronomy_173

如图所示, $O$ 处为地球, 卫星 1 环绕地球做匀速圆周运动, 卫星 2 环绕地球运行的轨道为椭圆, 两轨道不在同一平面内. 已知圆轨道的直径等于粗圆轨道的长轴, 且地球位于椭圆轨道的一个焦点上, 引力常量为 $G$ 、地球的质量为 $M$, 卫星 1 的轨道半径为 $R, O Q=1.5 R$.下列说法正确的是 ## 卫星 1 [图1] A: 卫星 1 的运行周期大于卫星 2 的运行周期 B: 如果卫星 1 的环绕速度为 $v$, 卫星 2 在 $Q$ 点的速度为 $v_{Q}$, 则 $v<v_{Q}$ C: 卫星 2 在 $Q$ 点的速度 $v_{Q}>\sqrt{\frac{2 G M}{3 R}}$ D: 如果卫星 1 的加速度为 $a$, 卫星 2 在 $P$ 点的加速度为 $a_{p}$, 则 $a<a_{p}$

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, $O$ 处为地球, 卫星 1 环绕地球做匀速圆周运动, 卫星 2 环绕地球运行的轨道为椭圆, 两轨道不在同一平面内. 已知圆轨道的直径等于粗圆轨道的长轴, 且地球位于椭圆轨道的一个焦点上, 引力常量为 $G$ 、地球的质量为 $M$, 卫星 1 的轨道半径为 $R, O Q=1.5 R$.下列说法正确的是 ## 卫星 1 [图1] A: 卫星 1 的运行周期大于卫星 2 的运行周期 B: 如果卫星 1 的环绕速度为 $v$, 卫星 2 在 $Q$ 点的速度为 $v_{Q}$, 则 $v<v_{Q}$ C: 卫星 2 在 $Q$ 点的速度 $v_{Q}>\sqrt{\frac{2 G M}{3 R}}$ D: 如果卫星 1 的加速度为 $a$, 卫星 2 在 $P$ 点的加速度为 $a_{p}$, 则 $a<a_{p}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在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_1139

In science fiction films the asteroid belt is typically portrayed as a region of the Solar System where the spacecraft needs to dodge and weave its way through many large asteroids that are rather close together. However, if this image were true then very few probes would be able to pass through the belt into the outer Solar System. [figure1] Figure 1 Artist conceptual illustration of the asteroid belt (left). Schematic of the Solar System with the asteroid belt between Mars and Jupiter (right). This question will look at the real distances between asteroids.c. Assuming this volume is uniformly filled by spherical rocky asteroids of average radius $R_{a v}$, derive a relationship between the average distance between asteroids, $d_{a v}$, and their radius $R_{a v}$, remembering to keep the total mass equal to $M_{\text {belt }}$.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is an expression. Here is some context information for this question, which might assist you in solving it: In science fiction films the asteroid belt is typically portrayed as a region of the Solar System where the spacecraft needs to dodge and weave its way through many large asteroids that are rather close together. However, if this image were true then very few probes would be able to pass through the belt into the outer Solar System. [figure1] Figure 1 Artist conceptual illustration of the asteroid belt (left). Schematic of the Solar System with the asteroid belt between Mars and Jupiter (right). This question will look at the real distances between asteroids. problem: c. Assuming this volume is uniformly filled by spherical rocky asteroids of average radius $R_{a v}$, derive a relationship between the average distance between asteroids, $d_{a v}$, and their radius $R_{a v}$, remembering to keep the total mass equal to $M_{\text {belt }}$. 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_626

已知地球的质量为 $M$, 半径为 $R$, 引力常量为 $G$ 。赤道上地球表面附近的重力加速度用 $g_{e}$ 表示, 北极处地球表面附近的重力加速度用 $g_{N}$ 表示, 将地球视为均匀球体。 用 $g_{e} 、 g_{N}$ 和半径 $R$ 表示地球同步卫星的轨道半径。 [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: 已知地球的质量为 $M$, 半径为 $R$, 引力常量为 $G$ 。赤道上地球表面附近的重力加速度用 $g_{e}$ 表示, 北极处地球表面附近的重力加速度用 $g_{N}$ 表示, 将地球视为均匀球体。 用 $g_{e} 、 g_{N}$ 和半径 $R$ 表示地球同步卫星的轨道半径。 [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_957

One possible theory for why the gas giants have ring systems is that a small moon got too close to the parent planet. When the gravitational tidal forces (due to the difference between the strength of the planet's pull on the near and far sides of the moon) became greater than the gravitational forces holding the moon together, it was ripped apart. This minimum distance is called the "Roche limit", named after the French astronomer Edouard Roche who first calculated it. It is defined as when the gravitational force generated by the moon at its surface is equal to the tidal forces it experiences at that distance. [figure1] Consider a spherical planet with mass $M$ and radius $R$, and a perfectly rigid spherical moon with mass $m$ and radius $r$, orbiting the planet in a circular orbit of radius $d$. For a small particle of mass $u$ on the surface of the moon, the gravitational and tidal forces it experiences will be $$ F_{\text {grav }}=\frac{G m u}{r^{2}} \quad F_{\text {tidal }}=\frac{2 G M u r}{d^{3}} $$ The inner edge of Saturn's rings ( $\mathrm{D}$ ring) occurs at $1.11 R_{\text {Saturn }}$, and the outer edge of the A ring (the last main visible ring) is at $2.27 R_{\text {Saturn }}$. Do the rings of Saturn fall (roughly) between the two Roche limits calculated for the extreme cases of a perfectly rigid and a fluid moon made of water ice?

You are participating in an international Astronomy competition and need to solve the following question. This is a True or False question. problem: One possible theory for why the gas giants have ring systems is that a small moon got too close to the parent planet. When the gravitational tidal forces (due to the difference between the strength of the planet's pull on the near and far sides of the moon) became greater than the gravitational forces holding the moon together, it was ripped apart. This minimum distance is called the "Roche limit", named after the French astronomer Edouard Roche who first calculated it. It is defined as when the gravitational force generated by the moon at its surface is equal to the tidal forces it experiences at that distance. [figure1] Consider a spherical planet with mass $M$ and radius $R$, and a perfectly rigid spherical moon with mass $m$ and radius $r$, orbiting the planet in a circular orbit of radius $d$. For a small particle of mass $u$ on the surface of the moon, the gravitational and tidal forces it experiences will be $$ F_{\text {grav }}=\frac{G m u}{r^{2}} \quad F_{\text {tidal }}=\frac{2 G M u r}{d^{3}} $$ The inner edge of Saturn's rings ( $\mathrm{D}$ ring) occurs at $1.11 R_{\text {Saturn }}$, and the outer edge of the A ring (the last main visible ring) is at $2.27 R_{\text {Saturn }}$. Do the rings of Saturn fall (roughly) between the two Roche limits calculated for the extreme cases of a perfectly rigid and a fluid moon made of water ice? You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be either "True" or "False".

TF

Astronomy

EN

multi-modal

Astronomy_831

A stable open cluster of about $N=1000$ sun-like stars has an angular diameter of $\theta=30$ arc minutes and distance of $d=500 \mathrm{pc}$. Assuming the cluster can be approximated by a sphere of uniform density, estimate the average velocities of stars in the cluster. The gravitational potential energy of a sphere of uniform density and radius $r$ is $$ U_{\text {sphere }}=-\frac{3}{5} \frac{G M_{\text {sphere }}^{2}}{r} $$ A: $507 \mathrm{~m} / \mathrm{s}$ B: $643 \mathrm{~m} / \mathrm{s}$ C: $894 \mathrm{~m} / \mathrm{s}$ D: $1021 \mathrm{~m} / \mathrm{s}$ E: $771 \mathrm{~m} / \mathrm{s}$

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: A stable open cluster of about $N=1000$ sun-like stars has an angular diameter of $\theta=30$ arc minutes and distance of $d=500 \mathrm{pc}$. Assuming the cluster can be approximated by a sphere of uniform density, estimate the average velocities of stars in the cluster. The gravitational potential energy of a sphere of uniform density and radius $r$ is $$ U_{\text {sphere }}=-\frac{3}{5} \frac{G M_{\text {sphere }}^{2}}{r} $$ A: $507 \mathrm{~m} / \mathrm{s}$ B: $643 \mathrm{~m} / \mathrm{s}$ C: $894 \mathrm{~m} / \mathrm{s}$ D: $1021 \mathrm{~m} / \mathrm{s}$ E: $771 \mathrm{~m} / \mathrm{s}$ You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D, E].

SC

Astronomy

EN

text-only

Astronomy_1195

In which of the following places is the length of the shortest day of the year equal to half the length of the longest night? A: Dubai $\left(\varphi=25^{\circ} \mathrm{N}\right)$ B: London $\left(\varphi=52^{\circ} \mathrm{N}\right)$ C: Rio de Janeiro $\left(\varphi=23^{\circ} \mathrm{S}\right)$ D: Troms√∏ $\left(\varphi=70^{\circ} \mathrm{N}\right)$

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: In which of the following places is the length of the shortest day of the year equal to half the length of the longest night? A: Dubai $\left(\varphi=25^{\circ} \mathrm{N}\right)$ B: London $\left(\varphi=52^{\circ} \mathrm{N}\right)$ C: Rio de Janeiro $\left(\varphi=23^{\circ} \mathrm{S}\right)$ D: Troms√∏ $\left(\varphi=70^{\circ} \mathrm{N}\right)$ 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_73

一颗人造卫星在地球表面附近做匀速圆周运动, 经过 $t$ 时间, 卫星运行的路程为 $s$,运动半径转过的角度为 $\theta$, 引力常量为 $G$, 则 ( ) A: 地球的半径为 $\frac{s}{\theta}$ B: 地球的质量为 $\frac{s^{2}}{G \theta t^{2}}$ C: 地球的密度为 $\frac{3 \theta^{2}}{4 \pi G t^{2}}$ D: 地球表面的重力加速度为 $\frac{s \theta}{t}$

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: 一颗人造卫星在地球表面附近做匀速圆周运动, 经过 $t$ 时间, 卫星运行的路程为 $s$,运动半径转过的角度为 $\theta$, 引力常量为 $G$, 则 ( ) A: 地球的半径为 $\frac{s}{\theta}$ B: 地球的质量为 $\frac{s^{2}}{G \theta t^{2}}$ C: 地球的密度为 $\frac{3 \theta^{2}}{4 \pi G t^{2}}$ D: 地球表面的重力加速度为 $\frac{s \theta}{t}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

text-only

Astronomy_949

One possible theory for why the gas giants have ring systems is that a small moon got too close to the parent planet. When the gravitational tidal forces (due to the difference between the strength of the planet's pull on the near and far sides of the moon) became greater than the gravitational forces holding the moon together, it was ripped apart. This minimum distance is called the "Roche limit", named after the French astronomer Edouard Roche who first calculated it. It is defined as when the gravitational force generated by the moon at its surface is equal to the tidal forces it experiences at that distance. [figure1] Consider a spherical planet with mass $M$ and radius $R$, and a perfectly rigid spherical moon with mass $m$ and radius $r$, orbiting the planet in a circular orbit of radius $d$. For a small particle of mass $u$ on the surface of the moon, the gravitational and tidal forces it experiences will be $$ F_{\text {grav }}=\frac{G m u}{r^{2}} \quad F_{\text {tidal }}=\frac{2 G M u r}{d^{3}} $$ In practice, as a moon approaches the Roche limit it will start to deform and become more of an ellipsoid than a sphere, causing the tidal forces to increase, and so the Roche limit from our simple model is really a minimum radius. The opposite extreme would be to assume that both the planet and moon are made of a fluid, and so can deform without resistance (this works well when looking at things like stars in close binary systems). In that situation it can be shown that the equivalent formula for the Roche limit becomes $$ d_{R L} \approx 2.44 R\left(\frac{\rho_{M}}{\rho_{m}}\right)^{\frac{1}{3}} $$ Work out this new maximum value for the Roche limit for water ice around Saturn.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. problem: One possible theory for why the gas giants have ring systems is that a small moon got too close to the parent planet. When the gravitational tidal forces (due to the difference between the strength of the planet's pull on the near and far sides of the moon) became greater than the gravitational forces holding the moon together, it was ripped apart. This minimum distance is called the "Roche limit", named after the French astronomer Edouard Roche who first calculated it. It is defined as when the gravitational force generated by the moon at its surface is equal to the tidal forces it experiences at that distance. [figure1] Consider a spherical planet with mass $M$ and radius $R$, and a perfectly rigid spherical moon with mass $m$ and radius $r$, orbiting the planet in a circular orbit of radius $d$. For a small particle of mass $u$ on the surface of the moon, the gravitational and tidal forces it experiences will be $$ F_{\text {grav }}=\frac{G m u}{r^{2}} \quad F_{\text {tidal }}=\frac{2 G M u r}{d^{3}} $$ In practice, as a moon approaches the Roche limit it will start to deform and become more of an ellipsoid than a sphere, causing the tidal forces to increase, and so the Roche limit from our simple model is really a minimum radius. The opposite extreme would be to assume that both the planet and moon are made of a fluid, and so can deform without resistance (this works well when looking at things like stars in close binary systems). In that situation it can be shown that the equivalent formula for the Roche limit becomes $$ d_{R L} \approx 2.44 R\left(\frac{\rho_{M}}{\rho_{m}}\right)^{\frac{1}{3}} $$ Work out this new maximum value for the Roche limit for water ice around Saturn. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of km, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

multi-modal

Astronomy_334

《流浪地球 2 》影片中, 太空电梯高箃入云, 在地表与太空间高速穿梭。已知地球半径为 $6371 \mathrm{~km}$, 不考虑地球自转. 当太空电梯上升到离地高度 $6371 \mathrm{~km}$ 时, 质量为 $2.5 \mathrm{~kg}$的物体受到的重力约为 ( ) [图1] A: $0 \mathrm{~N}$ B: $25 \mathrm{~N}$ C: $12.5 \mathrm{~N}$ D: $6.25 \mathrm{~N}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 《流浪地球 2 》影片中, 太空电梯高箃入云, 在地表与太空间高速穿梭。已知地球半径为 $6371 \mathrm{~km}$, 不考虑地球自转. 当太空电梯上升到离地高度 $6371 \mathrm{~km}$ 时, 质量为 $2.5 \mathrm{~kg}$的物体受到的重力约为 ( ) [图1] A: $0 \mathrm{~N}$ B: $25 \mathrm{~N}$ C: $12.5 \mathrm{~N}$ D: $6.25 \mathrm{~N}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在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_292

一颗赤道上空运行的人造地球卫星, 其轨道半径为 $2 \mathrm{R}$ ( $\mathrm{R}$ 为地球半径), 卫星的运动方向与地球的自转方向相同, 设地球的自转角速度为 $\omega_{0}$, 若此时该卫星正好在赤道某建筑物的正上方, 则到它下次再通过该建筑物上方所需时间为（）（地表重力加速度为 $\mathrm{g}$ ) A: $2 \pi \sqrt{\frac{2 R}{g}}$ B: $2 \pi /\left(\sqrt{\frac{g}{2 R}}-\omega_{0}\right)$ C: $4 \pi \sqrt{\frac{2 R}{g}}$ D: $2 \pi /\left(\sqrt{\frac{g}{8 R}}-\omega_{0}\right)$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 一颗赤道上空运行的人造地球卫星, 其轨道半径为 $2 \mathrm{R}$ ( $\mathrm{R}$ 为地球半径), 卫星的运动方向与地球的自转方向相同, 设地球的自转角速度为 $\omega_{0}$, 若此时该卫星正好在赤道某建筑物的正上方, 则到它下次再通过该建筑物上方所需时间为（）（地表重力加速度为 $\mathrm{g}$ ) A: $2 \pi \sqrt{\frac{2 R}{g}}$ B: $2 \pi /\left(\sqrt{\frac{g}{2 R}}-\omega_{0}\right)$ C: $4 \pi \sqrt{\frac{2 R}{g}}$ D: $2 \pi /\left(\sqrt{\frac{g}{8 R}}-\omega_{0}\right)$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

text-only

Astronomy_350

建立物理模型是解决实际问题的重要方法。 如图 1 所示, 圆和椭圆是分析卫星运动时常用的模型。已知, 地球质量为 $M$, 半径为 $R$, 万有引力常量为 $G$ 。 在科幻电影《流浪地球》中有这样一个场景: 地球在木星的强大引力作用下, 加速向木星靠近，当地球与木星球心之间的距离小于某个值 $d$ 时，地球表面物体就会被木星吸走，进而导致地球可能被撕裂。这个临界距离 $d$ 被称为“洛希极限”。已知，木星和地球的密度分别为 $\rho_{0}$ 和 $\rho$, 木星和地球的半径分别为 $R_{0}$ 和 $R$, 且 $d \gg R$ 。请据此近似推导木星使地球产生撕裂危险的临界距离 $d$ ——“洛希极限”的表达式。【提示: 当 $x$ 很小时, $(1+x)^{n} \approx 1+n x_{0}$ 】 图 1 [图2] [图2] 图 2

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 建立物理模型是解决实际问题的重要方法。 如图 1 所示, 圆和椭圆是分析卫星运动时常用的模型。已知, 地球质量为 $M$, 半径为 $R$, 万有引力常量为 $G$ 。 在科幻电影《流浪地球》中有这样一个场景: 地球在木星的强大引力作用下, 加速向木星靠近，当地球与木星球心之间的距离小于某个值 $d$ 时，地球表面物体就会被木星吸走，进而导致地球可能被撕裂。这个临界距离 $d$ 被称为“洛希极限”。已知，木星和地球的密度分别为 $\rho_{0}$ 和 $\rho$, 木星和地球的半径分别为 $R_{0}$ 和 $R$, 且 $d \gg R$ 。请据此近似推导木星使地球产生撕裂危险的临界距离 $d$ ——“洛希极限”的表达式。【提示: 当 $x$ 很小时, $(1+x)^{n} \approx 1+n x_{0}$ 】 图 1 [图2] [图2] 图 2 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

EX

Astronomy

ZH

multi-modal

Astronomy_1165

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

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: The Parker Solar Probe (PSP) is part of a mission to learn more about the Sun, named after the scientist that first proposed the existence of the solar wind, and was launched on $12^{\text {th }}$ August 2018. Over the course of the 7 year mission it will orbit the Sun 24 times, and through 7 flybys of Venus it will lose some energy in order to get into an ever tighter orbit (see Figure 1). In its final 3 orbits it will have a perihelion (closest approach to the Sun) of only $r_{\text {peri }}=9.86 R_{\odot}$, about 7 times closer than any previous probe, the first of which is due on $24^{\text {th }}$ December 2024. In this extreme environment the probe will not only face extreme brightness and temperatures but also will break the record for the fastest ever spacecraft. [figure1] Figure 1: Left: The journey PSP will take to get from the Earth to the final orbit around the Sun. Right: The probe just after assembly in the John Hopkins University Applied Physics Laboratory. Credit: NASA / John Hopkins APL / Ed Whitman. $$ v^{2}=G M\left(\frac{2}{r}-\frac{1}{a}\right) $$ Given that in its final orbit PSP has a orbital period of 88 days, calculate the speed of the probe as it passes through the minimum perihelion. Give your answer in $\mathrm{km} \mathrm{s}^{-1}$. Close to the Sun the communications equipment is very sensitive to the extreme environment, so the mission is planned for the probe to take all of its primary science measurements whilst within 0.25 au of the Sun, and then to spend the rest of the orbit beaming that data back to Earth, as shown in Figure 2. [figure2] Figure 2: The way PSP is planned to split each orbit into taking measurements and sending data back. Credit: NASA / Johns Hopkins APL. When considering the position of an object in an elliptical orbit as a function of time, there are two important angles (called 'anomalies') necessary to do the calculation, and they are defined in Figure 3. By constructing a circular orbit centred on the same point as the ellipse and with the same orbital period, the eccentric anomaly, $E$, is then the angle between the major axis and the perpendicular projection of the object (some time $t$ after perihelion) onto the circle as measured from the centre of the ellipse ( $\angle x c z$ in the figure). The mean anomaly, $M$, is the angle between the major axis and where the object would have been at time $t$ if it was indeed on the circular orbit ( $\angle y c z$ in the figure, such that the shaded areas are the same). [figure3] Figure 3: The definitions of the anomalies needed to get the position of an object in an ellipse as a function of time. The Sun (located at the focus) is labeled $S$ and the probe $P . M$ and $E$ are the mean and eccentric anomalies respectively. The angle $\theta$ is called the true anomaly and is not needed for this question. Credit: Wikipedia. The eccentric anomaly can be related to the mean anomaly through Kepler's Equation, $$ M=E-e \sin E \text {. } $$ problem: b. Given that in its final orbit PSP has an orbital period of 88 days, calculate the speed of the probe as it passes through the minimum perihelion. Give your answer in $\mathrm{km} \mathrm{s}^{-1}$. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of \mathrm{~km} \mathrm{~s}^{-1}, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

multi-modal

Astronomy_20

由中国科学院、中国工程院两院院士评出的 2012 年中国十大科技进展新闻, 于 2013 年 1 月 19 日揭晓, “神九”载人飞船与“天宫一号”成功对接和“蛟龙”号下潜突破 7000 米分别排在第一、第二。若地球半径为 $R$, 把地球看做质量分布均匀的球体。“蛟龙”下潜深度为 $d$, “天宫一号”轨道距离地面高度为 $h$, “蛟龙”号所在处与“天宫一号”所在处的加速度大小之比为 ( ) A: $\frac{R-d}{R+h}$ B: $\frac{(R-d)^{2}}{(R+h)^{2}}$ C: $\frac{(R-d)(R+h)^{2}}{R^{3}}$ D: $(R-d)(R+h)$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 由中国科学院、中国工程院两院院士评出的 2012 年中国十大科技进展新闻, 于 2013 年 1 月 19 日揭晓, “神九”载人飞船与“天宫一号”成功对接和“蛟龙”号下潜突破 7000 米分别排在第一、第二。若地球半径为 $R$, 把地球看做质量分布均匀的球体。“蛟龙”下潜深度为 $d$, “天宫一号”轨道距离地面高度为 $h$, “蛟龙”号所在处与“天宫一号”所在处的加速度大小之比为 ( ) A: $\frac{R-d}{R+h}$ B: $\frac{(R-d)^{2}}{(R+h)^{2}}$ C: $\frac{(R-d)(R+h)^{2}}{R^{3}}$ D: $(R-d)(R+h)$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

text-only

Astronomy_178

宇宙中, 两颗靠得比较近的恒星, 只受到彼此之间的万有引力作用互相绕转, 称之为双星系统, 设某双星系统绕其连线上的 $O$ 点做匀速圆周运动, 转动周期为 $T$, 轨道半径分别为 $R_{A} 、 R_{B}$ 且 $R_{A}<R_{B}$, 引力常量 $G$ 已知, 则下列说法正确的是（） [图1] A: 星球 A 所受的向心力大于星球 B 所受的向心力 B: 星球 $\mathrm{A}$ 的线速度一定等于星球 $\mathrm{B}$ 的线速度 C: 星球 $\mathrm{A}$ 和星球 $\mathrm{B}$ 的质量之和为 $\frac{4 \pi^{2}\left(R_{A}+R_{B}\right)}{G T^{2}}$ D: 双星的总质量一定, 双星之间的距离越大, 其转动周期越大

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: 宇宙中, 两颗靠得比较近的恒星, 只受到彼此之间的万有引力作用互相绕转, 称之为双星系统, 设某双星系统绕其连线上的 $O$ 点做匀速圆周运动, 转动周期为 $T$, 轨道半径分别为 $R_{A} 、 R_{B}$ 且 $R_{A}<R_{B}$, 引力常量 $G$ 已知, 则下列说法正确的是（） [图1] A: 星球 A 所受的向心力大于星球 B 所受的向心力 B: 星球 $\mathrm{A}$ 的线速度一定等于星球 $\mathrm{B}$ 的线速度 C: 星球 $\mathrm{A}$ 和星球 $\mathrm{B}$ 的质量之和为 $\frac{4 \pi^{2}\left(R_{A}+R_{B}\right)}{G T^{2}}$ D: 双星的总质量一定, 双星之间的距离越大, 其转动周期越大 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_1124

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}}$.b. The Starkiller Base is able to stop nuclear fusion in the Sun's core i. At its current luminosity, how long would it take the Sun to radiate away all of its gravitational binding energy? (This is an estimate of how long it would take to drain a whole star when radiatively charging the superweapon.)

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 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: b. The Starkiller Base is able to stop nuclear fusion in the Sun's core i. At its current luminosity, how long would it take the Sun to radiate away all of its gravitational binding energy? (This is an estimate of how long it would take to drain a whole star when radiatively charging the superweapon.) 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_768

The picture below shows a satellite galaxy of the Milky Way: What is the name of this satellite galaxy? [figure1] A: Sagittarius Dwarf Spheroidal B: Triangulum Galaxy C: Andromeda Dwarf D: Large Magellanic Cloud

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 picture below shows a satellite galaxy of the Milky Way: What is the name of this satellite galaxy? [figure1] A: Sagittarius Dwarf Spheroidal B: Triangulum Galaxy C: Andromeda Dwarf D: Large Magellanic Cloud 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_360

中国对火星探测不解追求, 火星与地球距离最近的时刻最适合登陆火星和在地面对火星进行观测。设定火星、地球绕太阳做匀速圆周运动的轨道在同一平面内, 火星绕太 阳运动的轨道半径是地球绕太阳运动的轨道半径的 $k$ 倍 $(k>1)$, 地球绕太阳运动的周期为 $T_{0}$ 。如图为某时刻火星与地球距离最近时的示意图, 则到火星与地球再次距离最近所需的最短时间为 ( ) [图1] A: $\frac{k^{\frac{3}{2}}}{k^{\frac{3}{2}}-1} T_{0}$ B: $\frac{k^{\frac{2}{3}}}{k^{\frac{2}{3}}-1} T_{0}$ C: $\frac{k^{\frac{3}{2}}+1}{k^{\frac{3}{2}}} T_{0}$ D: $\frac{k^{\frac{2}{3}}+1}{k^{\frac{2}{3}}} T_{0}$

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: 中国对火星探测不解追求, 火星与地球距离最近的时刻最适合登陆火星和在地面对火星进行观测。设定火星、地球绕太阳做匀速圆周运动的轨道在同一平面内, 火星绕太 阳运动的轨道半径是地球绕太阳运动的轨道半径的 $k$ 倍 $(k>1)$, 地球绕太阳运动的周期为 $T_{0}$ 。如图为某时刻火星与地球距离最近时的示意图, 则到火星与地球再次距离最近所需的最短时间为 ( ) [图1] A: $\frac{k^{\frac{3}{2}}}{k^{\frac{3}{2}}-1} T_{0}$ B: $\frac{k^{\frac{2}{3}}}{k^{\frac{2}{3}}-1} T_{0}$ C: $\frac{k^{\frac{3}{2}}+1}{k^{\frac{3}{2}}} T_{0}$ D: $\frac{k^{\frac{2}{3}}+1}{k^{\frac{2}{3}}} T_{0}$ 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]

SC

Astronomy

ZH

multi-modal

Astronomy_784

What is the correct numerical value and unit of the Boltzmann constant? A: $1.38 \times 10^{-21} \mathrm{~m}^{3} \cdot \mathrm{kg} \cdot \mathrm{s}^{-2} \cdot \mathrm{K}^{-1}$ B: $1.38 \times 10^{-22} \mathrm{~m}^{2} \cdot \mathrm{kg} \cdot \mathrm{s}^{-3} \cdot \mathrm{K}^{-1}$ C: $1.38 \times 10^{-23} \mathrm{~m}^{2} \cdot \mathrm{kg} \cdot \mathrm{s}^{-2} \cdot \mathrm{K}^{-1}$ D: $1.38 \times 10^{-24} \mathrm{~m}^{2} \cdot \mathrm{kg} \cdot \mathrm{s}^{-2} \cdot \mathrm{K}^{-2}$

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: What is the correct numerical value and unit of the Boltzmann constant? A: $1.38 \times 10^{-21} \mathrm{~m}^{3} \cdot \mathrm{kg} \cdot \mathrm{s}^{-2} \cdot \mathrm{K}^{-1}$ B: $1.38 \times 10^{-22} \mathrm{~m}^{2} \cdot \mathrm{kg} \cdot \mathrm{s}^{-3} \cdot \mathrm{K}^{-1}$ C: $1.38 \times 10^{-23} \mathrm{~m}^{2} \cdot \mathrm{kg} \cdot \mathrm{s}^{-2} \cdot \mathrm{K}^{-1}$ D: $1.38 \times 10^{-24} \mathrm{~m}^{2} \cdot \mathrm{kg} \cdot \mathrm{s}^{-2} \cdot \mathrm{K}^{-2}$ You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

SC

Astronomy

EN

text-only

Astronomy_219

我国天文学家通过 FAST, 在武仙座球状星团 M13 中发现一个脉冲双星系统。如图所示, 假设在太空中有恒星 $A 、 B$ 双星系统绕点 $O$ 做顺时针匀速圆周运动, 运动周期为 $T_{l}$, 它们的轨道半径分别为 $\mathrm{R}_{\mathrm{A}} 、 \mathrm{R}_{\mathrm{B}}, \mathrm{R}_{\mathrm{A}}<\mathrm{R}_{\mathrm{B}}, C$ 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, 且 $A$ 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 $G$, 则以下说法正确的是 ( ) [图1] A: 若知道 $C$ 的轨道半径, 则可求出 $C$ 的质量 B: 恒星 $B$ 的质量为 $M_{B}=\frac{4 \pi^{2} R_{B}\left(R_{A}+R_{B}\right)^{2}}{G T_{1}^{2}}$ C: 若 $A$ 也有一颗运动周期为 $T_{2}$ 的卫星, 则其轨道半径一定小于 $C$ 的轨道半径 D: 设 $A 、 B 、 C$ 三星由图示位置到再次共线的时间为 $t$, 则 $t=\frac{T_{1} T_{2}}{\left.4 T_{1}+T_{2}\right)}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 我国天文学家通过 FAST, 在武仙座球状星团 M13 中发现一个脉冲双星系统。如图所示, 假设在太空中有恒星 $A 、 B$ 双星系统绕点 $O$ 做顺时针匀速圆周运动, 运动周期为 $T_{l}$, 它们的轨道半径分别为 $\mathrm{R}_{\mathrm{A}} 、 \mathrm{R}_{\mathrm{B}}, \mathrm{R}_{\mathrm{A}}<\mathrm{R}_{\mathrm{B}}, C$ 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, 且 $A$ 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 $G$, 则以下说法正确的是 ( ) [图1] A: 若知道 $C$ 的轨道半径, 则可求出 $C$ 的质量 B: 恒星 $B$ 的质量为 $M_{B}=\frac{4 \pi^{2} R_{B}\left(R_{A}+R_{B}\right)^{2}}{G T_{1}^{2}}$ C: 若 $A$ 也有一颗运动周期为 $T_{2}$ 的卫星, 则其轨道半径一定小于 $C$ 的轨道半径 D: 设 $A 、 B 、 C$ 三星由图示位置到再次共线的时间为 $t$, 则 $t=\frac{T_{1} T_{2}}{\left.4 T_{1}+T_{2}\right)}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

multi-modal

Astronomy_1128

Wolf-Rayet (WR) stars are some of the hottest stars known, with very strong stellar winds causing considerable mass to the be lost to the interstellar medium (ISM). In binary systems between a WR star and a very large $\mathrm{O}$ or $\mathrm{B}$ spectral class star, where their strong stellar winds collide can create the conditions for the formation of dust which goes on to enrich the ISM. [figure1] Figure 3: Left: A view of the WR140 binary system taken with the James Webb Space Telescope (JWST) in July 2022, showing clearly at least 17 nested dust shells. Credit: NASA/ESA/CSA/STScI/JPL-Caltech. Right: A radial plot along the image in the three mid infrared JWST filters used corresponding to 7.7, 15 and $21 \mu \mathrm{m}$, as well as the model of the dust production. The peaks correspond to each shell. Shells 2 and 17 are indicated on the model and the median shell separation is shown by the grey vertical lines. The projected distance is given in arcseconds. Credit: Lau et al. (2022). The WR140 system consists of a WR and an O star which produce dust very regularly when the two stars are close together, around periastron. They are in a highly elliptical orbit $(e=0.8993)$ with a period of 2895 days. Once far from the stars, these dust shells move through space at a remarkably constant speed as indicated by the regularity of the shells in the recent image taken with the James Webb Space Telescope (JWST), shown above in Figure 3. An artist's impression of the two stars in the system and the orbit (in the reference frame of the WR star) is shown in Figure 4 below. [figure2] Figure 4: Left: The relative size of the Sun, upper left, compared to the two stars in the system WR140. The O-type star is $\sim 30 M_{\odot}$, while its companion is $\sim 10 M_{\odot}$. Credit: NASA/JPL-Caltech. Right: The projected orbital configuration of WR 140 in the reference frame of the WR star. The red solid region around the periastron passage is where the O star is when dust is being formed. Credit: Lau et al. (2022).a. Take the distance to the system to be $1.64 \mathrm{kpc}$. ii. Shell 1 was observed by JWST on 27 th July 2022 to be 1.63 " away from the central stars. It was formed during the last periastron passage of the $\mathrm{O}$ star, which (as viewed from Earth) took place in December 2016. Taking light travel time into account, in what year was the periastron passage responsible for shell 17?

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: Wolf-Rayet (WR) stars are some of the hottest stars known, with very strong stellar winds causing considerable mass to the be lost to the interstellar medium (ISM). In binary systems between a WR star and a very large $\mathrm{O}$ or $\mathrm{B}$ spectral class star, where their strong stellar winds collide can create the conditions for the formation of dust which goes on to enrich the ISM. [figure1] Figure 3: Left: A view of the WR140 binary system taken with the James Webb Space Telescope (JWST) in July 2022, showing clearly at least 17 nested dust shells. Credit: NASA/ESA/CSA/STScI/JPL-Caltech. Right: A radial plot along the image in the three mid infrared JWST filters used corresponding to 7.7, 15 and $21 \mu \mathrm{m}$, as well as the model of the dust production. The peaks correspond to each shell. Shells 2 and 17 are indicated on the model and the median shell separation is shown by the grey vertical lines. The projected distance is given in arcseconds. Credit: Lau et al. (2022). The WR140 system consists of a WR and an O star which produce dust very regularly when the two stars are close together, around periastron. They are in a highly elliptical orbit $(e=0.8993)$ with a period of 2895 days. Once far from the stars, these dust shells move through space at a remarkably constant speed as indicated by the regularity of the shells in the recent image taken with the James Webb Space Telescope (JWST), shown above in Figure 3. An artist's impression of the two stars in the system and the orbit (in the reference frame of the WR star) is shown in Figure 4 below. [figure2] Figure 4: Left: The relative size of the Sun, upper left, compared to the two stars in the system WR140. The O-type star is $\sim 30 M_{\odot}$, while its companion is $\sim 10 M_{\odot}$. Credit: NASA/JPL-Caltech. Right: The projected orbital configuration of WR 140 in the reference frame of the WR star. The red solid region around the periastron passage is where the O star is when dust is being formed. Credit: Lau et al. (2022). problem: a. Take the distance to the system to be $1.64 \mathrm{kpc}$. ii. Shell 1 was observed by JWST on 27 th July 2022 to be 1.63 " away from the central stars. It was formed during the last periastron passage of the $\mathrm{O}$ star, which (as viewed from Earth) took place in December 2016. Taking light travel time into account, in what year was the periastron passage responsible for shell 17? 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