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Astronomy_373

北斗卫星导航系统由地球同步静止轨道卫星 $a$ 、与地球自转周期相同的倾斜地球同步轨道卫星 $b$, 以及比它们轨道低一些的轨道卫星 $c$ 组成, 它们均为圆轨道卫星。若轨道卫星 $c$ 与地球同步静止轨道卫星 $a$ 在同一平面内同向旋转, 已知卫星 $c$ 的轨道半径为 $r$, 同步卫星轨道半径为 $4 r$, 地球自转周期为 $T$, 万有引力常量为 $G$, 下列说法正确的是 ( ) [图1] A: 卫星 $a$ 的发射速度大于地球第一宇宙速度, 轨道运行速度小于地球第一宇宙速度 B: 卫星 $a$ 与卫星 $b$ 具有相同的机械能 C: 地球的质量为 $\frac{4 \pi^{2} r^{3}}{G T^{2}}$ D: 卫星 $a$ 与卫星 $c$ 周期之比为 $8: 1$, 某时刻两者相距最近, 则经过 $\frac{T}{8}$ 后, 两者再次相距最近

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 北斗卫星导航系统由地球同步静止轨道卫星 $a$ 、与地球自转周期相同的倾斜地球同步轨道卫星 $b$, 以及比它们轨道低一些的轨道卫星 $c$ 组成, 它们均为圆轨道卫星。若轨道卫星 $c$ 与地球同步静止轨道卫星 $a$ 在同一平面内同向旋转, 已知卫星 $c$ 的轨道半径为 $r$, 同步卫星轨道半径为 $4 r$, 地球自转周期为 $T$, 万有引力常量为 $G$, 下列说法正确的是 ( ) [图1] A: 卫星 $a$ 的发射速度大于地球第一宇宙速度, 轨道运行速度小于地球第一宇宙速度 B: 卫星 $a$ 与卫星 $b$ 具有相同的机械能 C: 地球的质量为 $\frac{4 \pi^{2} r^{3}}{G T^{2}}$ D: 卫星 $a$ 与卫星 $c$ 周期之比为 $8: 1$, 某时刻两者相距最近, 则经过 $\frac{T}{8}$ 后, 两者再次相距最近 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

ZH

multi-modal

Astronomy_479

如果你想利用通过同步卫星转发信号的无线电话与对方通话, 你讲完电话后, 至少要等多长时间才能听到对方通话？（已知地球质量 $M=6.0 \times 10^{24} \mathrm{~kg}$, 地球半径 $R=6.4 \times 10^{6} \mathrm{~m}$, 引力常量 $G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2}$ )

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数值。 问题: 如果你想利用通过同步卫星转发信号的无线电话与对方通话, 你讲完电话后, 至少要等多长时间才能听到对方通话？（已知地球质量 $M=6.0 \times 10^{24} \mathrm{~kg}$, 地球半径 $R=6.4 \times 10^{6} \mathrm{~m}$, 引力常量 $G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2}$ ) 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。 请记住，你的答案应以s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_762

The picture below shows a very close and well-known galaxy: [figure1] What is the name of this galaxy? A: Pinwheel Galaxy B: Whirlpool Galaxy C: Bode Galaxy D: Triangulum Galaxy

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 very close and well-known galaxy: [figure1] What is the name of this galaxy? A: Pinwheel Galaxy B: Whirlpool Galaxy C: Bode Galaxy D: Triangulum Galaxy You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

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Astronomy

EN

multi-modal

Astronomy_868

Ben Chen is an alien living on a system identical to earth, except his planet's obliquity is $0^{\circ}$. Located at latitude $42.20^{\circ}$, he wants to observe M52. Due to the open cluster being so dim, Ben needs perfect conditions to observe M52. Due to atmospheric effects, M52 can only be observed above an altitude of $30^{\circ}$. Additionally, it must be during astronomical twilight (when the Sun is more than $18^{\circ}$ below the horizon). Of the following dates, which is the earliest after the vernal equinox that Ben can observe the cluster? The coordinates of M52 are approximately $\alpha=0 \mathrm{~h}$ and $\delta=60^{\circ}$. A: April 21st B: June 21st C: August 21st D: October 21st E: December 21st

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: Ben Chen is an alien living on a system identical to earth, except his planet's obliquity is $0^{\circ}$. Located at latitude $42.20^{\circ}$, he wants to observe M52. Due to the open cluster being so dim, Ben needs perfect conditions to observe M52. Due to atmospheric effects, M52 can only be observed above an altitude of $30^{\circ}$. Additionally, it must be during astronomical twilight (when the Sun is more than $18^{\circ}$ below the horizon). Of the following dates, which is the earliest after the vernal equinox that Ben can observe the cluster? The coordinates of M52 are approximately $\alpha=0 \mathrm{~h}$ and $\delta=60^{\circ}$. A: April 21st B: June 21st C: August 21st D: October 21st E: December 21st 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_696

2019 年 11 月 5 日, 第 49 颗北斗导航卫星成功发射, 为 2020 年完成北斗全球组网打下坚实基础。北斗卫星导航系统由不同轨道卫星组成, 其中北斗-IGSO3 卫星的运行轨道为倾斜地球同步轨道, 倾角为 $55.9^{\circ}$, 高度约为 3.59 万千米; 北斗-M3 卫星运行轨道为中地球轨道, 倾角为 $55.3^{\circ}$, 高度约为 2.16 万千米。已知地球半径约为 6400 千米,两颗卫星的运行轨道均可视为圆轨道，则下列说法中正确的是（） [图1] A: 北斗-IGSO3 卫星的线速度大于北斗-M3 卫星的线速度 B: 北斗-IGSO3 卫星的周期大于北斗-M3 卫星的周期 C: 北斗-IGSO3 卫星连续经过地球非赤道上某处正上方的时间间隔约为 $24 \mathrm{~h}$ D: 北斗-IGSO3 卫星与地面上的北京市的距离恒定

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 2019 年 11 月 5 日, 第 49 颗北斗导航卫星成功发射, 为 2020 年完成北斗全球组网打下坚实基础。北斗卫星导航系统由不同轨道卫星组成, 其中北斗-IGSO3 卫星的运行轨道为倾斜地球同步轨道, 倾角为 $55.9^{\circ}$, 高度约为 3.59 万千米; 北斗-M3 卫星运行轨道为中地球轨道, 倾角为 $55.3^{\circ}$, 高度约为 2.16 万千米。已知地球半径约为 6400 千米,两颗卫星的运行轨道均可视为圆轨道，则下列说法中正确的是（） [图1] A: 北斗-IGSO3 卫星的线速度大于北斗-M3 卫星的线速度 B: 北斗-IGSO3 卫星的周期大于北斗-M3 卫星的周期 C: 北斗-IGSO3 卫星连续经过地球非赤道上某处正上方的时间间隔约为 $24 \mathrm{~h}$ D: 北斗-IGSO3 卫星与地面上的北京市的距离恒定 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在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_644

如图所示, 某航天器围绕一颗半径为 $R$ 的行星做匀速圆周运动, 其环绕周期为 $T$,经过轨道上 $A$ 点时发出了一束激光, 与行星表面相切于 $B$ 点, 若测得激光束 $A B$ 与轨道半径 $A O$ 夹角为 $\theta$, 引力常量为 $G$, 不考虑行星的自转, 下列说法正确的是 ( ) [图1] A: 行星的质量为 $\frac{4 \pi^{2} R^{3}}{G T^{2} \sin ^{3} \theta}$ B: 行星的平均密度为 $\frac{3 \pi}{G T^{2} \sin ^{3} \theta}$ C: 行星表面的重力加速度为 $\frac{4 \pi^{2} R}{T^{2} \sin ^{3} \theta}$ D: 行星赤道表面随行星自转做匀速圆周运动的线速度为 $\frac{2 \pi R}{T \sin \theta}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 如图所示, 某航天器围绕一颗半径为 $R$ 的行星做匀速圆周运动, 其环绕周期为 $T$,经过轨道上 $A$ 点时发出了一束激光, 与行星表面相切于 $B$ 点, 若测得激光束 $A B$ 与轨道半径 $A O$ 夹角为 $\theta$, 引力常量为 $G$, 不考虑行星的自转, 下列说法正确的是 ( ) [图1] A: 行星的质量为 $\frac{4 \pi^{2} R^{3}}{G T^{2} \sin ^{3} \theta}$ B: 行星的平均密度为 $\frac{3 \pi}{G T^{2} \sin ^{3} \theta}$ C: 行星表面的重力加速度为 $\frac{4 \pi^{2} R}{T^{2} \sin ^{3} \theta}$ D: 行星赤道表面随行星自转做匀速圆周运动的线速度为 $\frac{2 \pi R}{T \sin \theta}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_1089

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

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

NV

Astronomy

EN

multi-modal

Astronomy_346

“开普勒 -47 ”, 该系统位于天鹅座内, 距离地球大约 5000 光年, 这一新的系统有一对互相围绕彼此运行的恒星，运行周期为 $T$, 其中一颗大恒星的质量为 $M$ ，另一颗小恒星质量只有大恒星质量的三分之一, 已知引力常量为 $G$, 则下列判断正确的是 ( ) A: 大恒星与小恒星的角速度之比为 $1: 1 B: 大恒星与小恒星的向心加速度之比为 1 C: 大恒星与小恒星相距 $\sqrt[3]{\frac{G M T^{2}}{3 \pi^{2}}}$ D: 小恒星的轨道半径为 $\frac{3}{4} \sqrt[3]{\frac{G M T^{2}}{3 \pi^{2}}}$

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: “开普勒 -47 ”, 该系统位于天鹅座内, 距离地球大约 5000 光年, 这一新的系统有一对互相围绕彼此运行的恒星，运行周期为 $T$, 其中一颗大恒星的质量为 $M$ ，另一颗小恒星质量只有大恒星质量的三分之一, 已知引力常量为 $G$, 则下列判断正确的是 ( ) A: 大恒星与小恒星的角速度之比为 $1: 1 B: 大恒星与小恒星的向心加速度之比为 1 C: 大恒星与小恒星相距 $\sqrt[3]{\frac{G M T^{2}}{3 \pi^{2}}}$ D: 小恒星的轨道半径为 $\frac{3}{4} \sqrt[3]{\frac{G M T^{2}}{3 \pi^{2}}}$ 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_105

有一颗中高轨道卫星在赤道上空自西向东绕地球做圆周运动, 其轨道半径为地球同步卫星轨道半径的四分之一。某时刻该卫星正好经过赤道上某建筑物, 已知同步卫星的周期为 $\mathrm{T}_{0}$ ，则下列说法正确的是（） A: 该卫星的周期为 $\frac{T_{0}}{4}$ B: 该卫星的向心力为同步卫星的 $\frac{1}{16}$ C: 再经 $\frac{T_{0}}{8}$ 的时间该卫星将再次经过该建筑物 D: 再经 $\frac{T_{0}}{7}$ 的时间该卫星将再次经过该建筑物

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 有一颗中高轨道卫星在赤道上空自西向东绕地球做圆周运动, 其轨道半径为地球同步卫星轨道半径的四分之一。某时刻该卫星正好经过赤道上某建筑物, 已知同步卫星的周期为 $\mathrm{T}_{0}$ ，则下列说法正确的是（） A: 该卫星的周期为 $\frac{T_{0}}{4}$ B: 该卫星的向心力为同步卫星的 $\frac{1}{16}$ C: 再经 $\frac{T_{0}}{8}$ 的时间该卫星将再次经过该建筑物 D: 再经 $\frac{T_{0}}{7}$ 的时间该卫星将再次经过该建筑物 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在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_634

我国对火星的首次探测任务将于 2020 年正式开始实施, 要实现对火星的形貌、土 壤、环境、大气等的探测, 研究火星上的水冰分布、物理场和内部结构等。现假想为研究火星上有关重力的实验, 在火星的表面附近做一个上拖实验, 将一个小球以某一初速度坚直向上抛出, 测得小球相邻两次经过抛出点上方 $h$ 处的时间间隔为 $t$, 作出 $t^{2}-h$ 图像如图所示, 已知火星的半径为 $R$, 引力常量为 $G$, 则火星的质量为 ( ) [图1] A: $\frac{4 a R^{2}}{G b}$ B: $\frac{4 b R^{2}}{G a}$ C: $\frac{8 a R^{2}}{G b}$ D: $\frac{8 b R^{2}}{G a}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 我国对火星的首次探测任务将于 2020 年正式开始实施, 要实现对火星的形貌、土 壤、环境、大气等的探测, 研究火星上的水冰分布、物理场和内部结构等。现假想为研究火星上有关重力的实验, 在火星的表面附近做一个上拖实验, 将一个小球以某一初速度坚直向上抛出, 测得小球相邻两次经过抛出点上方 $h$ 处的时间间隔为 $t$, 作出 $t^{2}-h$ 图像如图所示, 已知火星的半径为 $R$, 引力常量为 $G$, 则火星的质量为 ( ) [图1] A: $\frac{4 a R^{2}}{G b}$ B: $\frac{4 b R^{2}}{G a}$ C: $\frac{8 a R^{2}}{G b}$ D: $\frac{8 b R^{2}}{G a}$ 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应为以下选项之一：[A, B, C, D]

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