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Astronomy_842

The surface of the Sun exhibits differential rotation, with different rotational periods at different latitudes. We can measure this rotation speed using Doppler spectroscopy or by tracking the motion of sunspots. If the rotation speed of the Sun's surface at the equator is $2021 \mathrm{~m} / \mathrm{s}$, and at $60^{\circ}$ South is $809 \mathrm{~m} / \mathrm{s}$, how long would it take for a sunspot at the equator to do a full extra lap around the Sun compared to a sunspot at $60^{\circ}$ South? A: 6.2 days B: 25.0 days C: 31.2 days D: 41.7 days E: 126 days

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: The surface of the Sun exhibits differential rotation, with different rotational periods at different latitudes. We can measure this rotation speed using Doppler spectroscopy or by tracking the motion of sunspots. If the rotation speed of the Sun's surface at the equator is $2021 \mathrm{~m} / \mathrm{s}$, and at $60^{\circ}$ South is $809 \mathrm{~m} / \mathrm{s}$, how long would it take for a sunspot at the equator to do a full extra lap around the Sun compared to a sunspot at $60^{\circ}$ South? A: 6.2 days B: 25.0 days C: 31.2 days D: 41.7 days E: 126 days You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D, E].

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Astronomy

EN

text-only

Astronomy_399

2021 年 5 月, “天问一号”探测器成功在火星软着陆, 我国成为世界上第一个首次探测火星就实现“绕、落、巡”三项任务的国家。(2) 火星探测器在火星附近的 $A$ 点减速后, 被火星捕获进入了 1 号粗圆轨道, 紧接着在 $B$ 点进行了一次“远火点平面机动”, 俗称“侧手翻”, 即从与火星赤道平行的 1 号轨道,调整为经过火星两极的 2 号轨道, 将探测器绕火星飞行的路线从“横着绕”变成“坚着绕”, 从而实现对火星表面的全面扫描, 如图 2 所示。以火星为参考系, 质量为 $M_{1}$ 的探测器沿 1 号轨道到达 $B$ 点时速度为 $v_{1}$, 为了实现“侧手翻”, 此时启动发动机, 在极短的时间内喷出部分气体, 假设气体为一次性喷出, 喷气后探测器质量变为 $M_{2}$ 、速度变为与 $v_{1}$ 垂直的 $v_{2}$ 。求喷出气体速度 $u$ 的大小。[图1] 图1 [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: 2021 年 5 月, “天问一号”探测器成功在火星软着陆, 我国成为世界上第一个首次探测火星就实现“绕、落、巡”三项任务的国家。(2) 火星探测器在火星附近的 $A$ 点减速后, 被火星捕获进入了 1 号粗圆轨道, 紧接着在 $B$ 点进行了一次“远火点平面机动”, 俗称“侧手翻”, 即从与火星赤道平行的 1 号轨道,调整为经过火星两极的 2 号轨道, 将探测器绕火星飞行的路线从“横着绕”变成“坚着绕”, 从而实现对火星表面的全面扫描, 如图 2 所示。以火星为参考系, 质量为 $M_{1}$ 的探测器沿 1 号轨道到达 $B$ 点时速度为 $v_{1}$, 为了实现“侧手翻”, 此时启动发动机, 在极短的时间内喷出部分气体, 假设气体为一次性喷出, 喷气后探测器质量变为 $M_{2}$ 、速度变为与 $v_{1}$ 垂直的 $v_{2}$ 。求喷出气体速度 $u$ 的大小。[图1] 图1 [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_1158

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.b. Jupiter has a period of 4332.589 days and Saturn has a period of 10759.22 days (where 1 day $=24$ hours). Note: be careful as your calculations will be very sensitive to rounding errors. i. Calculate the time between great conjunctions as viewed from the centre of the Solar System (this will be equal to the average synodic period). Give your answer in years (where 1 year $=365.25$ days). [Hint: consider a reference frame rotating at the same rate as Jupiter.]

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: b. Jupiter has a period of 4332.589 days and Saturn has a period of 10759.22 days (where 1 day $=24$ hours). Note: be careful as your calculations will be very sensitive to rounding errors. i. Calculate the time between great conjunctions as viewed from the centre of the Solar System (this will be equal to the average synodic period). Give your answer in years (where 1 year $=365.25$ days). [Hint: consider a reference frame rotating at the same rate as Jupiter.] All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of \text { years }, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

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Astronomy

EN

multi-modal

Astronomy_1018

The word 'equinox' comes from the Latin for "equal night", and typically people assume it's the time of year when day and night are equal in length (i.e. 12 hours each), but that's not strictly true - in fact the centre of the solar disc spends 12 hours above the horizon (see Fig 1), but sunrise and sunset times are measured from when the very top of the Sun first appears, and when it finally disappears respectively. [figure1] Figure 1: The motion of the Sun during the 12 hours of an equinox. By working out the angular radius of the Sun as seen from the Earth, work out how many extra minutes of daylight you get from this effect.

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 word 'equinox' comes from the Latin for "equal night", and typically people assume it's the time of year when day and night are equal in length (i.e. 12 hours each), but that's not strictly true - in fact the centre of the solar disc spends 12 hours above the horizon (see Fig 1), but sunrise and sunset times are measured from when the very top of the Sun first appears, and when it finally disappears respectively. [figure1] Figure 1: The motion of the Sun during the 12 hours of an equinox. By working out the angular radius of the Sun as seen from the Earth, work out how many extra minutes of daylight you get from this effect. 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 mins, 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_541

如图所示, 有 $\mathrm{A} 、 \mathrm{~B}$ 两颗卫星绕地心 $O$ 做圆周运动, 运动方向相反。 $\mathrm{A}$ 卫星的周期为 $T_{1}, \mathrm{~B}$ 卫星的周期为 $T_{2}$, 在某一时刻两卫星相距最近, 则（引力常量为 $G$ （） [图1] A: 两卫星下一次相距最近需经过时间 $t=\frac{T_{1} T_{2}}{T_{1}+T_{2}}$ B: 两颗卫星的轨道半径之比为 $\sqrt[3]{\frac{T_{1}^{2}}{T_{2}^{2}}}$ C: 若已知两颗卫星相距最近时的距离, 可求出地球的密度 D: 若已知两颗卫星相距最近时的距离, 可求出地球表面的重力加速度

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 如图所示, 有 $\mathrm{A} 、 \mathrm{~B}$ 两颗卫星绕地心 $O$ 做圆周运动, 运动方向相反。 $\mathrm{A}$ 卫星的周期为 $T_{1}, \mathrm{~B}$ 卫星的周期为 $T_{2}$, 在某一时刻两卫星相距最近, 则（引力常量为 $G$ （） [图1] A: 两卫星下一次相距最近需经过时间 $t=\frac{T_{1} T_{2}}{T_{1}+T_{2}}$ B: 两颗卫星的轨道半径之比为 $\sqrt[3]{\frac{T_{1}^{2}}{T_{2}^{2}}}$ C: 若已知两颗卫星相距最近时的距离, 可求出地球的密度 D: 若已知两颗卫星相距最近时的距离, 可求出地球表面的重力加速度 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_1037

The word 'equinox' comes from the Latin for "equal night", and typically people assume it's the time of year when day and night are equal in length (i.e. 12 hours each), but that's not strictly true - in fact the centre of the solar disc spends 12 hours above the horizon (see Fig 1), but sunrise and sunset times are measured from when the very top of the Sun first appears, and when it finally disappears respectively. [figure1] Figure 1: The motion of the Sun during the 12 hours of an equinox. Since the atmosphere has a refractive index just slightly greater than 1, this has an influence too, meaning that when the centre of the solar disc appears to be just at the horizon, it is in fact $0.6^{\circ}$ below the horizon. Combining this with the previous effect, calculate the total number of minutes longer than 12 hours a day is on an equinox.

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 word 'equinox' comes from the Latin for "equal night", and typically people assume it's the time of year when day and night are equal in length (i.e. 12 hours each), but that's not strictly true - in fact the centre of the solar disc spends 12 hours above the horizon (see Fig 1), but sunrise and sunset times are measured from when the very top of the Sun first appears, and when it finally disappears respectively. [figure1] Figure 1: The motion of the Sun during the 12 hours of an equinox. Since the atmosphere has a refractive index just slightly greater than 1, this has an influence too, meaning that when the centre of the solar disc appears to be just at the horizon, it is in fact $0.6^{\circ}$ below the horizon. Combining this with the previous effect, calculate the total number of minutes longer than 12 hours a day is on an equinox. 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 mins, 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_1125

The James Webb Space Telescope (JWST) is an incredibly exciting next generation telescope that was successfully launched on $25^{\text {th }}$ December 2021 . Its mirror is approximately $6.5 \mathrm{~m}$ in diameter, much larger than the $2.4 \mathrm{~m}$ mirror of the Hubble Space Telescope (HST), and so it has far greater resolution and sensitivity. Whilst HST largely imaged in the visible, JWST will do most of its work in the nearand mid-infrared (NIR and MIR respectively). This will allow it to pick up heavily redshifted light, such as that from the first generation of stars in the very first galaxies. [figure1] Figure 5: Left: A full-scale model of JWST next to some of the scientists and engineers involved in its development at the Goddard Space Flight Center. Credit: NASA / Goddard Space Flight Center / Pat Izzo. Right: The position of the second Lagrangian point, $L_{2}$, relative to the Earth. Credit: ESA. The resolution limit of a telescope is set by the amount of diffraction light rays experience as they enter the system, and is related to the diameter of a telescope, $D$, and the wavelength being observed, $\lambda$. The resolution limit of a CCD is set by the size of the pixels. Three of the imaging cameras on JWST are tabulated with some properties below: | Instrument | Wavelength range $(\mu \mathrm{m})$ | CCD plate scale (arcseconds / pixel) | | :---: | :---: | :---: | | NIRCam (short wave) | $0.6-2.3$ | 0.031 | | NIRCam (long wave) | $2.4-5.0$ | 0.065 | | MIRI | $5.6-25.5$ | 0.11 | An arcsecond is a measure of angle where $1^{\circ}=3600$ arcseconds. The familiar variation in intensity on a screen, $I_{\text {slit }}$, due to diffraction through an infinitely tall single slit is given as $$ I_{\text {slit }}=I_{0}\left(\frac{\sin (x)}{x}\right)^{2}, \text { where } \quad x=\frac{\pi D \theta}{\lambda} $$ and $I_{0}$ is the initial intensity. For a circular aperture, the formula is slightly different and is given as $$ I_{\mathrm{circ}}=I_{0}\left(\frac{2 J_{1}(x)}{x}\right)^{2} . $$ Here $J_{1}(x)$ is the Bessel function of the first kind and is calculated as $$ J_{n}(x)=\sum_{r=0}^{\infty} \frac{(-1)^{r}}{r !(n+r) !}\left(\frac{x}{2}\right)^{n+2 r} \quad \text { so } \quad J_{1}(x)=\frac{x}{2}\left(1-\frac{x^{2}}{8}+\frac{x^{4}}{192}-\ldots\right) . $$ The $x$-axis intercepts and shape of the maxima are quite different, as shown in Figure 6. The position of the first minimum of $I_{\text {slit }}$ is at $x_{\min }=\pi$ meaning that $\theta_{\min , \text { slit }}=\lambda / D$, whilst for $I_{\text {circ }}$ it is at $x_{\min }=3.8317 \ldots$ so $\theta_{\min , \mathrm{circ}} \approx 1.22 \lambda / D$. This is one way of defining the minimum angular resolution, although since the flux drops off so steeply away from the central maximum a more convenient one for use with CCDs is the angle corresponding to the full width half maximum (FWHM). [figure2] Figure 6: Left: The $I_{\text {slit }}$ (purple) and $I_{\text {circ }}$ (blue - the wider central maximum) functions, normalised so that $I_{0}=1$. You can see the shapes and $x$-intercepts are different. Right: How $x_{\min }$ and the full width half maximum (FWHM) are defined. Here it is shown for $I_{\text {circ }}$. As well as having the largest mirror of any space telescope ever launched, it is also one of the most sensitive, with its greatest sensitivity in the NIRCam F200W filter (centred on a wavelength of $1.989 \mu \mathrm{m})$ where after $10^{4}$ seconds it can detect a flux of $9.1 \mathrm{nJy}\left(1 \mathrm{Jy}=10^{-26} \mathrm{~W} \mathrm{~m}^{-2} \mathrm{~Hz}^{-1}\right.$ ) with a signal-to-noise ratio (S/N) of 10 , corresponding to an apparent magnitude of $m=29.0$. This extraordinary sensitivity can be used to pick up light from the earliest galaxies in the Universe. The scale factor, $a$, parameterises the expansion of the Universe since the Big Bang, and is related to the redshift, $z$, as $$ a=(1+z)^{-1} \quad \text { where } \quad z \equiv \frac{\lambda_{\text {obs }}-\lambda_{\mathrm{emit}}}{\lambda_{\mathrm{emit}}} $$ with $\lambda_{\text {obs }}$ the observed wavelength and $\lambda_{\text {emit }}$ the rest frame wavelength. The current rate of expansion of the Universe is given by the Hubble constant, $H_{0}$, and this is related to the current Hubble time, $t_{\mathrm{H}_{0}}$, and current Hubble distance, $D_{\mathrm{H}_{0}}$, as $$ t_{\mathrm{H}_{0}} \equiv H_{0}^{-1} \quad \text { and } \quad D_{\mathrm{H}_{0}} \equiv c t_{\mathrm{H}_{0}} \text {. } $$ Here the subscript 0 indicates the values are measured today. The Hubble constant is more appropriately known as the Hubble parameter as it is a function of time, and the evolution of $H$ as a function of $z$ is $$ E(z)=\frac{H}{H_{0}} \equiv\left[\Omega_{0, m}(1+z)^{3}+\Omega_{0, \Lambda}+\Omega_{0, r}(1+z)^{4}\right]^{1 / 2}, $$ where $\Omega$ is the normalised density parameter, and the subscript $m, r$, and $\Lambda$ indicate the contribution to $\Omega$ from matter, radiation, and dark energy, respectively. The proper age of the Universe $t(z)$ at redshift $z$ is best evaluated in terms of $a$ as $$ t=t_{\mathrm{H}_{0}} \int_{0}^{(1+z)^{-1}} \frac{a}{\left(\Omega_{0, m} a+\Omega_{0, \Lambda} a^{4}+\Omega_{0, r}\right)^{1 / 2}} \mathrm{~d} a $$ If $\Omega_{0, r}=0$ and $\Omega_{0, m}+\Omega_{0, \Lambda}=1$ (corresponding to what it known as a flat Universe), then via the standard integral $\int\left(b^{2}+x^{2}\right)^{-1 / 2} \mathrm{~d} x=\ln \left(x+\sqrt{b^{2}+x^{2}}\right)+C$ this integral can be evaluated analytically to give $$ t=t_{\mathrm{H}_{0}} \frac{2}{3 \Omega_{0, \Lambda}^{1 / 2}} \ln \left[\left(\frac{\Omega_{0, \Lambda}}{\Omega_{0, m}}\right)^{1 / 2}(1+z)^{-3 / 2}+\left(\frac{\Omega_{0, \Lambda}}{\Omega_{0, m}(1+z)^{3}}+1\right)^{1 / 2}\right] $$ Finally, the luminosity distance, $D_{L}(z)$, corresponding to the distance away that an object appears to be due to its measured flux given its intrinsic luminosity (i.e. $f \equiv L / 4 \pi D_{L}^{2}$ ) is given as $$ D_{L}=\left(1+z_{i}\right) D_{\mathrm{H}_{0}} \int_{0}^{z_{i}} \frac{1}{E(z)} \mathrm{d} z=\left(1+z_{i}\right) D_{\mathrm{H}_{0}} \int_{a_{i}}^{1} \frac{1}{\left(\Omega_{0, m} a+\Omega_{0, \Lambda} a^{4}+\Omega_{0, r}\right)^{1 / 2}} \mathrm{~d} a $$ where $z_{i}$ is the redshift of interest and $a_{i}$ is the equivalent scale factor. Even for the flat Universe case with $\Omega_{0, r}=0$ this integral cannot be be done analytically so must be evaluated numerically.b. To achieve suitable sampling, an image will be considered diffraction limited when it has $\geq 2$ pixels per $\theta_{\text {FWHM. }}$. The diameter of the JWST primary mirror is $6.5 \mathrm{~m}$, however since it is composed of hexagons and hexagonal in shape, it is not straightforward to work out the equivalent circular mirror diameter. To a good approximation it can be taken to be $6.0 \mathrm{~m}$. i. Given $\theta_{F W H M}=\alpha \lambda / D$, find $\alpha$ for $I_{\text {circ, }}$, giving your answer to 3 s.f.

You are participating in an international Astronomy competition and need to solve the following question. This question has more than one correct answer, you need to include them all. Here is some context information for this question, which might assist you in solving it: The James Webb Space Telescope (JWST) is an incredibly exciting next generation telescope that was successfully launched on $25^{\text {th }}$ December 2021 . Its mirror is approximately $6.5 \mathrm{~m}$ in diameter, much larger than the $2.4 \mathrm{~m}$ mirror of the Hubble Space Telescope (HST), and so it has far greater resolution and sensitivity. Whilst HST largely imaged in the visible, JWST will do most of its work in the nearand mid-infrared (NIR and MIR respectively). This will allow it to pick up heavily redshifted light, such as that from the first generation of stars in the very first galaxies. [figure1] Figure 5: Left: A full-scale model of JWST next to some of the scientists and engineers involved in its development at the Goddard Space Flight Center. Credit: NASA / Goddard Space Flight Center / Pat Izzo. Right: The position of the second Lagrangian point, $L_{2}$, relative to the Earth. Credit: ESA. The resolution limit of a telescope is set by the amount of diffraction light rays experience as they enter the system, and is related to the diameter of a telescope, $D$, and the wavelength being observed, $\lambda$. The resolution limit of a CCD is set by the size of the pixels. Three of the imaging cameras on JWST are tabulated with some properties below: | Instrument | Wavelength range $(\mu \mathrm{m})$ | CCD plate scale (arcseconds / pixel) | | :---: | :---: | :---: | | NIRCam (short wave) | $0.6-2.3$ | 0.031 | | NIRCam (long wave) | $2.4-5.0$ | 0.065 | | MIRI | $5.6-25.5$ | 0.11 | An arcsecond is a measure of angle where $1^{\circ}=3600$ arcseconds. The familiar variation in intensity on a screen, $I_{\text {slit }}$, due to diffraction through an infinitely tall single slit is given as $$ I_{\text {slit }}=I_{0}\left(\frac{\sin (x)}{x}\right)^{2}, \text { where } \quad x=\frac{\pi D \theta}{\lambda} $$ and $I_{0}$ is the initial intensity. For a circular aperture, the formula is slightly different and is given as $$ I_{\mathrm{circ}}=I_{0}\left(\frac{2 J_{1}(x)}{x}\right)^{2} . $$ Here $J_{1}(x)$ is the Bessel function of the first kind and is calculated as $$ J_{n}(x)=\sum_{r=0}^{\infty} \frac{(-1)^{r}}{r !(n+r) !}\left(\frac{x}{2}\right)^{n+2 r} \quad \text { so } \quad J_{1}(x)=\frac{x}{2}\left(1-\frac{x^{2}}{8}+\frac{x^{4}}{192}-\ldots\right) . $$ The $x$-axis intercepts and shape of the maxima are quite different, as shown in Figure 6. The position of the first minimum of $I_{\text {slit }}$ is at $x_{\min }=\pi$ meaning that $\theta_{\min , \text { slit }}=\lambda / D$, whilst for $I_{\text {circ }}$ it is at $x_{\min }=3.8317 \ldots$ so $\theta_{\min , \mathrm{circ}} \approx 1.22 \lambda / D$. This is one way of defining the minimum angular resolution, although since the flux drops off so steeply away from the central maximum a more convenient one for use with CCDs is the angle corresponding to the full width half maximum (FWHM). [figure2] Figure 6: Left: The $I_{\text {slit }}$ (purple) and $I_{\text {circ }}$ (blue - the wider central maximum) functions, normalised so that $I_{0}=1$. You can see the shapes and $x$-intercepts are different. Right: How $x_{\min }$ and the full width half maximum (FWHM) are defined. Here it is shown for $I_{\text {circ }}$. As well as having the largest mirror of any space telescope ever launched, it is also one of the most sensitive, with its greatest sensitivity in the NIRCam F200W filter (centred on a wavelength of $1.989 \mu \mathrm{m})$ where after $10^{4}$ seconds it can detect a flux of $9.1 \mathrm{nJy}\left(1 \mathrm{Jy}=10^{-26} \mathrm{~W} \mathrm{~m}^{-2} \mathrm{~Hz}^{-1}\right.$ ) with a signal-to-noise ratio (S/N) of 10 , corresponding to an apparent magnitude of $m=29.0$. This extraordinary sensitivity can be used to pick up light from the earliest galaxies in the Universe. The scale factor, $a$, parameterises the expansion of the Universe since the Big Bang, and is related to the redshift, $z$, as $$ a=(1+z)^{-1} \quad \text { where } \quad z \equiv \frac{\lambda_{\text {obs }}-\lambda_{\mathrm{emit}}}{\lambda_{\mathrm{emit}}} $$ with $\lambda_{\text {obs }}$ the observed wavelength and $\lambda_{\text {emit }}$ the rest frame wavelength. The current rate of expansion of the Universe is given by the Hubble constant, $H_{0}$, and this is related to the current Hubble time, $t_{\mathrm{H}_{0}}$, and current Hubble distance, $D_{\mathrm{H}_{0}}$, as $$ t_{\mathrm{H}_{0}} \equiv H_{0}^{-1} \quad \text { and } \quad D_{\mathrm{H}_{0}} \equiv c t_{\mathrm{H}_{0}} \text {. } $$ Here the subscript 0 indicates the values are measured today. The Hubble constant is more appropriately known as the Hubble parameter as it is a function of time, and the evolution of $H$ as a function of $z$ is $$ E(z)=\frac{H}{H_{0}} \equiv\left[\Omega_{0, m}(1+z)^{3}+\Omega_{0, \Lambda}+\Omega_{0, r}(1+z)^{4}\right]^{1 / 2}, $$ where $\Omega$ is the normalised density parameter, and the subscript $m, r$, and $\Lambda$ indicate the contribution to $\Omega$ from matter, radiation, and dark energy, respectively. The proper age of the Universe $t(z)$ at redshift $z$ is best evaluated in terms of $a$ as $$ t=t_{\mathrm{H}_{0}} \int_{0}^{(1+z)^{-1}} \frac{a}{\left(\Omega_{0, m} a+\Omega_{0, \Lambda} a^{4}+\Omega_{0, r}\right)^{1 / 2}} \mathrm{~d} a $$ If $\Omega_{0, r}=0$ and $\Omega_{0, m}+\Omega_{0, \Lambda}=1$ (corresponding to what it known as a flat Universe), then via the standard integral $\int\left(b^{2}+x^{2}\right)^{-1 / 2} \mathrm{~d} x=\ln \left(x+\sqrt{b^{2}+x^{2}}\right)+C$ this integral can be evaluated analytically to give $$ t=t_{\mathrm{H}_{0}} \frac{2}{3 \Omega_{0, \Lambda}^{1 / 2}} \ln \left[\left(\frac{\Omega_{0, \Lambda}}{\Omega_{0, m}}\right)^{1 / 2}(1+z)^{-3 / 2}+\left(\frac{\Omega_{0, \Lambda}}{\Omega_{0, m}(1+z)^{3}}+1\right)^{1 / 2}\right] $$ Finally, the luminosity distance, $D_{L}(z)$, corresponding to the distance away that an object appears to be due to its measured flux given its intrinsic luminosity (i.e. $f \equiv L / 4 \pi D_{L}^{2}$ ) is given as $$ D_{L}=\left(1+z_{i}\right) D_{\mathrm{H}_{0}} \int_{0}^{z_{i}} \frac{1}{E(z)} \mathrm{d} z=\left(1+z_{i}\right) D_{\mathrm{H}_{0}} \int_{a_{i}}^{1} \frac{1}{\left(\Omega_{0, m} a+\Omega_{0, \Lambda} a^{4}+\Omega_{0, r}\right)^{1 / 2}} \mathrm{~d} a $$ where $z_{i}$ is the redshift of interest and $a_{i}$ is the equivalent scale factor. Even for the flat Universe case with $\Omega_{0, r}=0$ this integral cannot be be done analytically so must be evaluated numerically. problem: b. To achieve suitable sampling, an image will be considered diffraction limited when it has $\geq 2$ pixels per $\theta_{\text {FWHM. }}$. The diameter of the JWST primary mirror is $6.5 \mathrm{~m}$, however since it is composed of hexagons and hexagonal in shape, it is not straightforward to work out the equivalent circular mirror diameter. To a good approximation it can be taken to be $6.0 \mathrm{~m}$. i. Given $\theta_{F W H M}=\alpha \lambda / D$, find $\alpha$ for $I_{\text {circ, }}$, giving your answer to 3 s.f. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Their answer types are, in order, [numerical value, numerical value]. Please end your response with: "The final answers are \boxed{ANSWER}", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5

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2020 年 12 月 17 日“嫦娥五号”首次地外天体采样返回任务圆满完成。在采样返回过程中, “嫦娥五号”要面对取样、上升、对接和高速再入等四个主要技术难题, 要进行多次变轨飞行。“嫦娥五号”绕月球飞行的三条轨道示意图如图所示, 轨道 1 是贴近月球表面的圆形轨道, 轨道 2 和轨道 3 是变轨后的椭圆轨道, 并且都与轨道 1 相切于 $A$ 点。 $A$点是轨道 2 的近月点, $B$ 点是轨道 2 的远月点, “嫦娥五号”在轨道 1 上的运行速率约为 $1.7 \mathrm{~km} / \mathrm{s}$ 。不计变轨中“嫦娥五号”的质量变化, 不考虑其他天体的影响, 下列说法中正确试卷第 62 页，共 107 页 的是 ( ) [图1] A: “嫦娥五号”在轨道 2 经过 $A$ 点时的加速度大于在轨道 1 经过 $A$ 点时的加速度 B: “嫦娥五号”在轨道 2 经过 $B$ 点时的速率一定小于 $1.7 \mathrm{~km} / \mathrm{s}$ C: “嫦娥五号”在轨道 3 上运行的最大速率小于其在轨道 2 上运行的最大速率 D: “嫦娥五号”在轨道 3 所具有的机械能小于其在轨道 2 所具有的机械能

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: 2020 年 12 月 17 日“嫦娥五号”首次地外天体采样返回任务圆满完成。在采样返回过程中, “嫦娥五号”要面对取样、上升、对接和高速再入等四个主要技术难题, 要进行多次变轨飞行。“嫦娥五号”绕月球飞行的三条轨道示意图如图所示, 轨道 1 是贴近月球表面的圆形轨道, 轨道 2 和轨道 3 是变轨后的椭圆轨道, 并且都与轨道 1 相切于 $A$ 点。 $A$点是轨道 2 的近月点, $B$ 点是轨道 2 的远月点, “嫦娥五号”在轨道 1 上的运行速率约为 $1.7 \mathrm{~km} / \mathrm{s}$ 。不计变轨中“嫦娥五号”的质量变化, 不考虑其他天体的影响, 下列说法中正确试卷第 62 页，共 107 页 的是 ( ) [图1] A: “嫦娥五号”在轨道 2 经过 $A$ 点时的加速度大于在轨道 1 经过 $A$ 点时的加速度 B: “嫦娥五号”在轨道 2 经过 $B$ 点时的速率一定小于 $1.7 \mathrm{~km} / \mathrm{s}$ C: “嫦娥五号”在轨道 3 上运行的最大速率小于其在轨道 2 上运行的最大速率 D: “嫦娥五号”在轨道 3 所具有的机械能小于其在轨道 2 所具有的机械能 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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2021 年 5 月, “天问一号”探测器成功在火星软着陆, 我国成为世界上第一个首次探测火星就实现“绕、落、巡”三项任务的国家。火星探测器在火星附近的 $A$ 点减速后, 被火星捕获进入了 1 号粗圆轨道, 紧接着在 $B$ 点进行了一次“远火点平面机动”, 俗称“侧手翻”, 即从与火星赤道平行的 1 号轨道,调整为经过火星两极的 2 号轨道, 将探测器绕火星飞行的路线从“横着绕”变成“坚着绕”, 从而实现对火星表面的全面扫描, 如图 2 所示。以火星为参考系, 质量为 $M_{1}$ 的探测器沿 1 号轨道到达 $B$ 点时速度为 $v_{1}$, 为了实现“侧手翻”, 此时启动发动机, 在极短的时间内喷出部分气体, 假设气体为一次性喷出, 喷气后探测器质量变为 $M_{2}$ 、速度变为与 $v_{1}$ 垂直的 $v_{2}$ 。 假设实现“侧手翻”的能量全部来源于化学能, 化学能向动能转化比例为 $k(k<1)$, 求此次“侧手翻”消耗的化学能 $\Delta E$ 。[图1] 图1 [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: 2021 年 5 月, “天问一号”探测器成功在火星软着陆, 我国成为世界上第一个首次探测火星就实现“绕、落、巡”三项任务的国家。火星探测器在火星附近的 $A$ 点减速后, 被火星捕获进入了 1 号粗圆轨道, 紧接着在 $B$ 点进行了一次“远火点平面机动”, 俗称“侧手翻”, 即从与火星赤道平行的 1 号轨道,调整为经过火星两极的 2 号轨道, 将探测器绕火星飞行的路线从“横着绕”变成“坚着绕”, 从而实现对火星表面的全面扫描, 如图 2 所示。以火星为参考系, 质量为 $M_{1}$ 的探测器沿 1 号轨道到达 $B$ 点时速度为 $v_{1}$, 为了实现“侧手翻”, 此时启动发动机, 在极短的时间内喷出部分气体, 假设气体为一次性喷出, 喷气后探测器质量变为 $M_{2}$ 、速度变为与 $v_{1}$ 垂直的 $v_{2}$ 。 假设实现“侧手翻”的能量全部来源于化学能, 化学能向动能转化比例为 $k(k<1)$, 求此次“侧手翻”消耗的化学能 $\Delta E$ 。[图1] 图1 [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

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Astronomy_765

An object emits light at a wavelength of 200 nanometers. You receive the light at 1000 nanometers. What is the redshift $z$ ? A: $z=1.2$ B: $z=4$ C: $z=5$ D: $z=6$

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 object emits light at a wavelength of 200 nanometers. You receive the light at 1000 nanometers. What is the redshift $z$ ? A: $z=1.2$ B: $z=4$ C: $z=5$ D: $z=6$ 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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在圆轨道上运动的质量为 $m$ 的人造地球卫星, 它们到地面的距离等于地球半径 $R$,地面上的重力加速度为 $g$, 则 A: 卫星运动的速度为 $\sqrt{2 R g}$ B: 卫星运动的加速度为 $\frac{g}{2}$ C: 卫星运动的周期为 $4 \pi \sqrt{\frac{2 R}{g}}$ D: 卫星的动能为 $\frac{m g R}{4}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 在圆轨道上运动的质量为 $m$ 的人造地球卫星, 它们到地面的距离等于地球半径 $R$,地面上的重力加速度为 $g$, 则 A: 卫星运动的速度为 $\sqrt{2 R g}$ B: 卫星运动的加速度为 $\frac{g}{2}$ C: 卫星运动的周期为 $4 \pi \sqrt{\frac{2 R}{g}}$ D: 卫星的动能为 $\frac{m g R}{4}$ 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应为两个或更多的选项：[A, B, C, D]

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19 世纪末, 有科学家提出了太空电梯的构想: 在赤道上建设一座直到地球同步卫星轨道的高塔, 并在塔内架设电梯。这种电梯可用于发射人造卫星, 其发射方法是将卫星通过太空电梯缓慢地提升到预定轨道高度处, 然后再启动推进装置将卫星从太空电梯发射出去, 使其直接进入预定圆轨道。已知地球质量为 $M$ 、半径为 $R$ 、自转周期为 $T$,万有引力常量为 $G$ 。 求高塔的高度 $h_{0}$;

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 19 世纪末, 有科学家提出了太空电梯的构想: 在赤道上建设一座直到地球同步卫星轨道的高塔, 并在塔内架设电梯。这种电梯可用于发射人造卫星, 其发射方法是将卫星通过太空电梯缓慢地提升到预定轨道高度处, 然后再启动推进装置将卫星从太空电梯发射出去, 使其直接进入预定圆轨道。已知地球质量为 $M$ 、半径为 $R$ 、自转周期为 $T$,万有引力常量为 $G$ 。 求高塔的高度 $h_{0}$; 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_1111

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.f). Calculate the gravitational force of the supermoon on the Earth. What mass increase would a Moon at apogee need, to create the same gravitational force?

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: f). Calculate the gravitational force of the supermoon on the Earth. What mass increase would a Moon at apogee need, to create the same gravitational force? 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_{\mathcal{D}}, 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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已知人造航天器在某行星表面上空绕行星做匀速圆周运动, 绕行方向与行星自转方向相同 (人造航天器周期小于行星的自转周期), 经过时间 $t$ ( $t$ 小于航天器的绕行周期）, 航天器运动的弧长为 $s$, 航天器与行星的中心连线扫过角度为 $\theta$, 引力常量为 $G$, 航天器上的人相邻两次看到行星赤道上的标志物的时间间隔是 $\Delta t$, 这个行星的同步卫星的离行星的球心距离( ) A: $\frac{s \Delta t}{(2 \pi t-\theta \Delta t)}$ B: $\frac{s \Delta t}{(\theta \Delta t-2 \pi t)}$ C: $\frac{s}{\theta} \sqrt[3]{\frac{\theta^{2} \Delta t^{3}}{(2 \pi t-\theta \Delta t)^{2}}}$ D: $\frac{s}{\theta} \sqrt[3]{\frac{\theta^{2} \Delta t^{2}}{(\theta \Delta t-2 \pi t)^{2}}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 已知人造航天器在某行星表面上空绕行星做匀速圆周运动, 绕行方向与行星自转方向相同 (人造航天器周期小于行星的自转周期), 经过时间 $t$ ( $t$ 小于航天器的绕行周期）, 航天器运动的弧长为 $s$, 航天器与行星的中心连线扫过角度为 $\theta$, 引力常量为 $G$, 航天器上的人相邻两次看到行星赤道上的标志物的时间间隔是 $\Delta t$, 这个行星的同步卫星的离行星的球心距离( ) A: $\frac{s \Delta t}{(2 \pi t-\theta \Delta t)}$ B: $\frac{s \Delta t}{(\theta \Delta t-2 \pi t)}$ C: $\frac{s}{\theta} \sqrt[3]{\frac{\theta^{2} \Delta t^{3}}{(2 \pi t-\theta \Delta t)^{2}}}$ D: $\frac{s}{\theta} \sqrt[3]{\frac{\theta^{2} \Delta t^{2}}{(\theta \Delta t-2 \pi t)^{2}}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy_24

牛顿为证明地球对苹果的作用力和地球对月球的作用力是同一种性质的力, 遵循相同规律, 进行了“月-地检验”。已知月球的轨道半径约为地球半径的 60 倍, 则“月-地检验”是计算月球公转的 ( ) A: 线速度是地球自转地表线速度的 60 倍 B: 线速度是地球自转地表线速度的 $60^{2}$ 倍 C: 向心加速度是自由落体加速度的 $\frac{1}{60^{2}}$ 倍 D: 周期是地球自转周期的 $\frac{1}{60^{2}}$ 倍

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 牛顿为证明地球对苹果的作用力和地球对月球的作用力是同一种性质的力, 遵循相同规律, 进行了“月-地检验”。已知月球的轨道半径约为地球半径的 60 倍, 则“月-地检验”是计算月球公转的 ( ) A: 线速度是地球自转地表线速度的 60 倍 B: 线速度是地球自转地表线速度的 $60^{2}$ 倍 C: 向心加速度是自由落体加速度的 $\frac{1}{60^{2}}$ 倍 D: 周期是地球自转周期的 $\frac{1}{60^{2}}$ 倍 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

ZH

text-only

Astronomy_420

在星球 $M$ 上将一轻弹簧坚直固定在水平桌面上, 把物体 $P$ 轻放在弹簧上端, $P$ 由静止向下运动, 物体的加速度 $a$ 与弹簧的压缩量 $x$ 间的关系如图中实线所示. 在另一星球 $N$ 上用完全相同的弹簧, 改用物体 $Q$ 完成同样的过程, 其 $a-x$ 关系如图中虚线所示,假设两星球均为质量均匀分布的球体. 已知星球 $M$ 的半径是星球 $N$ 的 3 倍, 则 ( ) [图1] A: $M$ 与 $N$ 的密度相等 B: $Q$ 的质量是 $P$ 的 6 倍 C: $Q$ 下落过程中的最大动能是 $P$ 的 3 倍 D: $Q$ 下落过程中弹簧的最大压缩量是 $P$ 的 4 倍

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 在星球 $M$ 上将一轻弹簧坚直固定在水平桌面上, 把物体 $P$ 轻放在弹簧上端, $P$ 由静止向下运动, 物体的加速度 $a$ 与弹簧的压缩量 $x$ 间的关系如图中实线所示. 在另一星球 $N$ 上用完全相同的弹簧, 改用物体 $Q$ 完成同样的过程, 其 $a-x$ 关系如图中虚线所示,假设两星球均为质量均匀分布的球体. 已知星球 $M$ 的半径是星球 $N$ 的 3 倍, 则 ( ) [图1] A: $M$ 与 $N$ 的密度相等 B: $Q$ 的质量是 $P$ 的 6 倍 C: $Q$ 下落过程中的最大动能是 $P$ 的 3 倍 D: $Q$ 下落过程中弹簧的最大压缩量是 $P$ 的 4 倍 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_127

如图所示, 是地球大气层外圆形轨道上运动的三颗卫星, 且 $b$ 和 $c$ 在同一轨道上,则下列说法正确的是( ) [图1] A: $b 、 c$ 的周期相同，且大于 $a$ 的周期 B: $b 、 c$ 的线速度大小相等，且小于 $a$ 的线速度 C: $b$ 加速后可以实现与 $c$ 对接 D: $a$ 的线速度一定等于第一宇宙速度

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 如图所示, 是地球大气层外圆形轨道上运动的三颗卫星, 且 $b$ 和 $c$ 在同一轨道上,则下列说法正确的是( ) [图1] A: $b 、 c$ 的周期相同，且大于 $a$ 的周期 B: $b 、 c$ 的线速度大小相等，且小于 $a$ 的线速度 C: $b$ 加速后可以实现与 $c$ 对接 D: $a$ 的线速度一定等于第一宇宙速度 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_848

Where and when should we place a radio telescope such that, when combined with a radio telescope on Earth, the system could "see" the supermassive black hole in Sculptor's Galaxy (NGC $253)$ ? Sculptor's Galaxy's supermassive black hole's mass is estimated to be around $5 \cdot 10^{6} M_{\odot}$, and its distance is estimated to be around $3.5 \mathrm{Mpc}$. Out of the options below, pick the one closest to the estimate you obtain, rounding up. Consider the energy of a radio wave to be around $10^{-5} \mathrm{eV}$. Use the following formula to estimate the angular resolution: $\theta=\frac{\lambda}{D}$ A: On the Moon when it is at its apogee. B: On Mars when it is in conjunction. C: On Venus when it is in its greatest elongation. D: On one of Jupiter's moons when it is in opposition. E: Somewhere in the farthest points of the Oort cloud when Earth is at its perihelion.

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: Where and when should we place a radio telescope such that, when combined with a radio telescope on Earth, the system could "see" the supermassive black hole in Sculptor's Galaxy (NGC $253)$ ? Sculptor's Galaxy's supermassive black hole's mass is estimated to be around $5 \cdot 10^{6} M_{\odot}$, and its distance is estimated to be around $3.5 \mathrm{Mpc}$. Out of the options below, pick the one closest to the estimate you obtain, rounding up. Consider the energy of a radio wave to be around $10^{-5} \mathrm{eV}$. Use the following formula to estimate the angular resolution: $\theta=\frac{\lambda}{D}$ A: On the Moon when it is at its apogee. B: On Mars when it is in conjunction. C: On Venus when it is in its greatest elongation. D: On one of Jupiter's moons when it is in opposition. E: Somewhere in the farthest points of the Oort cloud when Earth is at its perihelion. 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_1021

In January 2020, NASA's Transiting Exoplanet Survey Satellite (TESS) discovered an Earth-sized exoplanet, called TOI $700 \mathrm{~d}$, in its star's habitable zone. This is the range of distances a planet can orbit a star so that liquid water can exist on the surface, given sufficient atmospheric pressure. It was discovered using the transit method, where the planet passes directly between the observer and the star, causing a drop in brightness. [figure1] Figure 1: The three planets of the TOI 700 system, illustrated here, orbit a small, cool M dwarf star. TOI $700 \mathrm{~d}$ is the first Earth-size habitable-zone world discovered by TESS. Credit: NASA's Goddard Space Flight Center. TOI $700 \mathrm{~d}$ has radius $R_{P}=1.19 R_{E}$ orbiting a star with luminosity $0.0233 L_{\odot}$ at a distance of 0.163 au. Assume that the planet absorbs all the light that hits the surface, and that the orbit is circular. (a) Determine the total power incident on the surface of the planet, $L_{\text {incident }}$. Assuming the planet is in thermal equilibrium and a perfect blackbody emitter with temperature $T_{P}$, the total amount of energy emitted is given by the Stefan-Boltzmann Law, $$ L_{\text {emitted }}=4 \pi R_{P}^{2} \sigma T_{P}^{4} $$ where $\sigma=5.67 \times 10^{-8} \mathrm{~W} \mathrm{~m}^{-2} \mathrm{~K}^{-4}$.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. problem: In January 2020, NASA's Transiting Exoplanet Survey Satellite (TESS) discovered an Earth-sized exoplanet, called TOI $700 \mathrm{~d}$, in its star's habitable zone. This is the range of distances a planet can orbit a star so that liquid water can exist on the surface, given sufficient atmospheric pressure. It was discovered using the transit method, where the planet passes directly between the observer and the star, causing a drop in brightness. [figure1] Figure 1: The three planets of the TOI 700 system, illustrated here, orbit a small, cool M dwarf star. TOI $700 \mathrm{~d}$ is the first Earth-size habitable-zone world discovered by TESS. Credit: NASA's Goddard Space Flight Center. TOI $700 \mathrm{~d}$ has radius $R_{P}=1.19 R_{E}$ orbiting a star with luminosity $0.0233 L_{\odot}$ at a distance of 0.163 au. Assume that the planet absorbs all the light that hits the surface, and that the orbit is circular. (a) Determine the total power incident on the surface of the planet, $L_{\text {incident }}$. Assuming the planet is in thermal equilibrium and a perfect blackbody emitter with temperature $T_{P}$, the total amount of energy emitted is given by the Stefan-Boltzmann Law, $$ L_{\text {emitted }}=4 \pi R_{P}^{2} \sigma T_{P}^{4} $$ where $\sigma=5.67 \times 10^{-8} \mathrm{~W} \mathrm{~m}^{-2} \mathrm{~K}^{-4}$. 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 KW, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

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Astronomy

EN

multi-modal

Astronomy_431

宇宙中存在一些质量相等且离其他恒星较远的四颗星组成的四星系统, 通常可忽略其他星体对它们的引力作用, 如图所示, 设四星系统中每个星体的质量均为 $m$, 半径均为 $R$, 四颗星稳定分布在边长为 $L$ 的正方形的四个顶点上、已知引力常量为 $G$, 关于四星系统，下列说法正确的是 [图1] A: 四颗星的向心加速度的大小均为 $\frac{2 \sqrt{2} G m}{L^{2}}$ B: 四颗星运行的线速度大小均为 $\frac{1}{2} \sqrt{\frac{(4+\sqrt{2}) G m}{L}}$ C: 四颗星运行的角速度大小均为 $\frac{1}{L} \sqrt{\frac{(1+2 \sqrt{2}) G m}{L}}$ D: 四颗星运行的周期均为 $2 \pi L \sqrt{\frac{2 L}{(1+2 \sqrt{2}) G m}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 宇宙中存在一些质量相等且离其他恒星较远的四颗星组成的四星系统, 通常可忽略其他星体对它们的引力作用, 如图所示, 设四星系统中每个星体的质量均为 $m$, 半径均为 $R$, 四颗星稳定分布在边长为 $L$ 的正方形的四个顶点上、已知引力常量为 $G$, 关于四星系统，下列说法正确的是 [图1] A: 四颗星的向心加速度的大小均为 $\frac{2 \sqrt{2} G m}{L^{2}}$ B: 四颗星运行的线速度大小均为 $\frac{1}{2} \sqrt{\frac{(4+\sqrt{2}) G m}{L}}$ C: 四颗星运行的角速度大小均为 $\frac{1}{L} \sqrt{\frac{(1+2 \sqrt{2}) G m}{L}}$ D: 四颗星运行的周期均为 $2 \pi L \sqrt{\frac{2 L}{(1+2 \sqrt{2}) G m}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

ZH

multi-modal

Astronomy_565

地球赤道上有一个观察者 $a$, 赤道平面内有一颗自西向东运行的近地卫星 $\mathrm{b}, a$ 观测发现, 其正上方有一颗静止不动的卫星 $\mathrm{c}$, 每隔时间 $T$ 卫星 $\mathrm{b}$ 就会从其正上方飞过,已知地球半径为 $R$, 地表处重力加速度为 $g$, 万有引力常量为 $G$, 下列说法正确的是 ( ) A: $\mathrm{c}$ 的加速度大于 $\mathrm{b}$ 的加速度 B: $a$ 的线速度大于 $\mathrm{c}$ 的线速度 C: 地球的质量为 $\frac{4 \pi^{2} R^{3}}{G T^{2}}$ D: $\mathrm{c}$ 的周期为 $\frac{2 \pi T \sqrt{R}}{T \sqrt{g}-2 \pi \sqrt{R}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 地球赤道上有一个观察者 $a$, 赤道平面内有一颗自西向东运行的近地卫星 $\mathrm{b}, a$ 观测发现, 其正上方有一颗静止不动的卫星 $\mathrm{c}$, 每隔时间 $T$ 卫星 $\mathrm{b}$ 就会从其正上方飞过,已知地球半径为 $R$, 地表处重力加速度为 $g$, 万有引力常量为 $G$, 下列说法正确的是 ( ) A: $\mathrm{c}$ 的加速度大于 $\mathrm{b}$ 的加速度 B: $a$ 的线速度大于 $\mathrm{c}$ 的线速度 C: 地球的质量为 $\frac{4 \pi^{2} R^{3}}{G T^{2}}$ D: $\mathrm{c}$ 的周期为 $\frac{2 \pi T \sqrt{R}}{T \sqrt{g}-2 \pi \sqrt{R}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

ZH

text-only

Astronomy_571

中国计划已经实现返回式月球软着陆器对月球进行科学探测, 如图所示, 发射一颗运动半径为 $r$ 的绕月卫星, 登月着陆器从绕月卫星出发 (不影响绕月卫星运动), 沿粗圆轨道降落到月球的表面上, 与月球表面经多次碰撞和弹跳才停下来。假设着陆器第一次弹起的最大高度为 $h$, 水平速度为 $v_{1}$, 第二次着陆时速度为 $v_{2}$, 已知月球半径为 $R$,着陆器质量为 $m$, 不计一切阻力和月球的自转。求: 在月球表面发射一颗月球卫星的最小发射速度是多大? [图1]

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 中国计划已经实现返回式月球软着陆器对月球进行科学探测, 如图所示, 发射一颗运动半径为 $r$ 的绕月卫星, 登月着陆器从绕月卫星出发 (不影响绕月卫星运动), 沿粗圆轨道降落到月球的表面上, 与月球表面经多次碰撞和弹跳才停下来。假设着陆器第一次弹起的最大高度为 $h$, 水平速度为 $v_{1}$, 第二次着陆时速度为 $v_{2}$, 已知月球半径为 $R$,着陆器质量为 $m$, 不计一切阻力和月球的自转。求: 在月球表面发射一颗月球卫星的最小发射速度是多大? [图1] 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

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Astronomy

ZH

multi-modal

Astronomy_318

压强表示单位面积上压力的大小, 是物理学中的重要概念。 理论上可以证明: 质量分布均匀的球壳对壳内物体的万有引力为零。利用该规律可给出一种计算恒星中心压强的模型: 恒星内部的热核反应会向外辐射大量的电磁波, 当辐射所产生的扩张压力与万有引力所产生的收缩压力平衡时, 恒星便稳定下来。 设想处于稳定状态的恒星是一质量分布均匀、密度为 $\rho$ 、半径为 $R$ 的球体。选取该恒星内部一距恒星中心为 $r(r \leq R)$ 、厚度为 $\Delta r(\Delta r$ 远小于 $r)$ 的小薄片 $A$, 如图所示, 已知辐射所产生的扩张压力在 $A$ 的内、外表面引起的压强差的绝对值为 $\triangle p$, 引力常量为 $G \circ$忽略其它天体的影响。 若恒星表面处扩张压力所产生的压强为零, 求恒星中心处的压强 $p_{C}$ 。 [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: 压强表示单位面积上压力的大小, 是物理学中的重要概念。 理论上可以证明: 质量分布均匀的球壳对壳内物体的万有引力为零。利用该规律可给出一种计算恒星中心压强的模型: 恒星内部的热核反应会向外辐射大量的电磁波, 当辐射所产生的扩张压力与万有引力所产生的收缩压力平衡时, 恒星便稳定下来。 设想处于稳定状态的恒星是一质量分布均匀、密度为 $\rho$ 、半径为 $R$ 的球体。选取该恒星内部一距恒星中心为 $r(r \leq R)$ 、厚度为 $\Delta r(\Delta r$ 远小于 $r)$ 的小薄片 $A$, 如图所示, 已知辐射所产生的扩张压力在 $A$ 的内、外表面引起的压强差的绝对值为 $\triangle p$, 引力常量为 $G \circ$忽略其它天体的影响。 若恒星表面处扩张压力所产生的压强为零, 求恒星中心处的压强 $p_{C}$ 。 [图1] 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

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Astronomy

ZH

multi-modal

Astronomy_921

The cosmic microwave background (CMB) is measured today to have an almost perfectly uniform temperature of $T_{0}=2.725 \mathrm{~K}$. The temperature of the CMB at any redshift can be calculated using $T=T_{0}(1+z)$ where $z$ is the redshift. During the early expansion of the Universe, the temperature was high enough to ionize the hydrogen atoms filling the universe and make the universe opaque. Once the temperature dropped below $3000 \mathrm{~K}$ the universe became transparent and the CMB was emitted in a process known as photon decoupling. [figure1] Figure 2: Left: The CMB has been measured to have an almost perfect black-body spectrum. Credit: NASA/WMAP Science Team. Right: The Planck satellite was launched to measure the tiny deviations from a perfect black-body. This is the map of the sky from their final data release from July 2018. Credit: ESA/Planck Collaboration. The relative size of the Universe, called the scale factor $a$, varies with redshift as $a=a_{0}(1+z)^{-1}$, whilst in a matter dominated universe (a valid assumption after the CMB was emitted) the scale factor varies with time $t$ as $a \propto t^{2 / 3}$. What was the temperature of the CMB when the Earth formed 4.5 Gyr ago?

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 cosmic microwave background (CMB) is measured today to have an almost perfectly uniform temperature of $T_{0}=2.725 \mathrm{~K}$. The temperature of the CMB at any redshift can be calculated using $T=T_{0}(1+z)$ where $z$ is the redshift. During the early expansion of the Universe, the temperature was high enough to ionize the hydrogen atoms filling the universe and make the universe opaque. Once the temperature dropped below $3000 \mathrm{~K}$ the universe became transparent and the CMB was emitted in a process known as photon decoupling. [figure1] Figure 2: Left: The CMB has been measured to have an almost perfect black-body spectrum. Credit: NASA/WMAP Science Team. Right: The Planck satellite was launched to measure the tiny deviations from a perfect black-body. This is the map of the sky from their final data release from July 2018. Credit: ESA/Planck Collaboration. The relative size of the Universe, called the scale factor $a$, varies with redshift as $a=a_{0}(1+z)^{-1}$, whilst in a matter dominated universe (a valid assumption after the CMB was emitted) the scale factor varies with time $t$ as $a \propto t^{2 / 3}$. What was the temperature of the CMB when the Earth formed 4.5 Gyr ago? 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 K, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

multi-modal

Astronomy_1147

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 $q=0.251 \rho X^{2} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \mathrm{~W} \mathrm{~kg}^{-1}$.c. Considering the evaluated equations for $\tau, R$, and $q$ we can use this with the measured luminosity of the Sun to get a new estimate for the central temperature. ii. Using this new central temperature, and considering $\mathrm{R}$ and the central number density of protons, $\mathrm{n}_{p}$, estimate the typical amount of time a proton needs to wait to undergo fusion, giving your answer in years.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: The 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: c. Considering the evaluated equations for $\tau, R$, and $q$ we can use this with the measured luminosity of the Sun to get a new estimate for the central temperature. ii. Using this new central temperature, and considering $\mathrm{R}$ and the central number density of protons, $\mathrm{n}_{p}$, estimate the typical amount of time a proton needs to wait to undergo fusion, giving your answer in years. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of \mathrm{yrs}, 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_8

北京时间 2023 年 5 月 10 日 21 时 22 分, 搭载天舟六号货运飞船的长征七号遥七运载火箭, 在我国海南文昌航天发射场成功点火发射, 预计将飞船送入近地点高度为 $200 \mathrm{~km}$ 、远地点高度为 $344 \mathrm{~km}$ 的近地轨道 (记为轨道I), 之后飞船再通过多次变轨完成与空间站的交会对接。已知空间站的运行轨道距地面的高度为 $400 \mathrm{~km}$ 。下列说法正确的是 ( ) A: 飞船在轨道I上近地点的加速度小于在远地点的加速度 B: 飞船由轨道I向高轨道变轨时, 需要减速 C: 飞船在轨道I上的运行周期小于空间站的运行周期 D: 对接完成后, 飞船运行的角速度小于地球自转的角速度

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 北京时间 2023 年 5 月 10 日 21 时 22 分, 搭载天舟六号货运飞船的长征七号遥七运载火箭, 在我国海南文昌航天发射场成功点火发射, 预计将飞船送入近地点高度为 $200 \mathrm{~km}$ 、远地点高度为 $344 \mathrm{~km}$ 的近地轨道 (记为轨道I), 之后飞船再通过多次变轨完成与空间站的交会对接。已知空间站的运行轨道距地面的高度为 $400 \mathrm{~km}$ 。下列说法正确的是 ( ) A: 飞船在轨道I上近地点的加速度小于在远地点的加速度 B: 飞船由轨道I向高轨道变轨时, 需要减速 C: 飞船在轨道I上的运行周期小于空间站的运行周期 D: 对接完成后, 飞船运行的角速度小于地球自转的角速度 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

text-only

Astronomy_511

一名宇航员登陆某星球, 对其进行探测。宇航员来到位于赤道的一平整的斜坡前,将一小球自斜坡底端正上方的 $O$ 点以初速度 $v_{0}$ 水平抛出, 如图所示。小球垂直击中了斜坡上的 $P$ 点, $P$ 点距水平地面的高度为 $h$ 。求: 该星球赤道地面处的重力加速度; [图1]

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 一名宇航员登陆某星球, 对其进行探测。宇航员来到位于赤道的一平整的斜坡前,将一小球自斜坡底端正上方的 $O$ 点以初速度 $v_{0}$ 水平抛出, 如图所示。小球垂直击中了斜坡上的 $P$ 点, $P$ 点距水平地面的高度为 $h$ 。求: 该星球赤道地面处的重力加速度; [图1] 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

EX

Astronomy

ZH

multi-modal

Astronomy_253

电影《流浪地球》讲述的是面对太阳快速老化膨胀的灾难, 人类制定了“流浪地球”计划, 这首先需要使自转角速度大小为 $\omega$ 的地球停止自转, 再将地球推移出太阳系到达距离太阳最近的恒星 (比邻星)。为了使地球停止自转, 设想的方案就是在地球赤道上均匀地安装 $N$ 台“喷气”发动机, 如下图所示 ( $\mathrm{N}$ 较大, 图中只画出了 4 个)。假设每台发动机均能沿赤道的切线方向提供大小恒为 $F$ 的推力, 该推力可阻碍地球的自转。已知描述地球转动的动力学方程与描述质点运动的牛顿第二定律方程 $F=m a$ 具有相似性, 为 $M=I \beta$, 其中 $M$ 为外力的总力矩, 即外力与对应力臂乘积的总和, 其值为 $N F R ; I$ 为地球相对地轴的转动惯量; $\beta$ 为单位时间内地球的角速度的改变量。将地球看成质量分布均匀的球体，下列说法中正确的是（） [图1] A: 在 $M=I \beta$ 与 $F=m a$ 的类比中, 与质量 $m$ 对应的物理量是转动惯量 $I$, 其物理意义是反映改变地球绕地轴转动情况的难易程度 B: $\beta$ 的单位为 $\mathrm{rad} / \mathrm{s}$ C: 地球自转刹车过程中, 赤道表面附近的重力加速度逐渐变小 D: 地球停止自转后, 赤道附近比两极点附近的重力加速度大

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: 电影《流浪地球》讲述的是面对太阳快速老化膨胀的灾难, 人类制定了“流浪地球”计划, 这首先需要使自转角速度大小为 $\omega$ 的地球停止自转, 再将地球推移出太阳系到达距离太阳最近的恒星 (比邻星)。为了使地球停止自转, 设想的方案就是在地球赤道上均匀地安装 $N$ 台“喷气”发动机, 如下图所示 ( $\mathrm{N}$ 较大, 图中只画出了 4 个)。假设每台发动机均能沿赤道的切线方向提供大小恒为 $F$ 的推力, 该推力可阻碍地球的自转。已知描述地球转动的动力学方程与描述质点运动的牛顿第二定律方程 $F=m a$ 具有相似性, 为 $M=I \beta$, 其中 $M$ 为外力的总力矩, 即外力与对应力臂乘积的总和, 其值为 $N F R ; I$ 为地球相对地轴的转动惯量; $\beta$ 为单位时间内地球的角速度的改变量。将地球看成质量分布均匀的球体，下列说法中正确的是（） [图1] A: 在 $M=I \beta$ 与 $F=m a$ 的类比中, 与质量 $m$ 对应的物理量是转动惯量 $I$, 其物理意义是反映改变地球绕地轴转动情况的难易程度 B: $\beta$ 的单位为 $\mathrm{rad} / \mathrm{s}$ C: 地球自转刹车过程中, 赤道表面附近的重力加速度逐渐变小 D: 地球停止自转后, 赤道附近比两极点附近的重力加速度大 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_96

两个星球组成双星, 它们在相互之间的万有引力作用下, 绕连线上某点做周期相同的匀速圆周运动, 现测得两星中心距离为 $\mathrm{R}$, 其运动周期为 $\mathrm{T}$, 若万有引力恒量为 $\mathrm{G}$,则两星的总质量为 ( ) A: $\frac{4 \pi^{2} R^{3}}{G T^{2}}$ B: $\frac{4 \pi^{2} R^{2}}{G T^{2}}$ C: $\frac{4 \pi^{2} R^{2}}{G T^{3}}$ D: $\frac{4 \pi^{2} R^{3}}{G T^{3}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 两个星球组成双星, 它们在相互之间的万有引力作用下, 绕连线上某点做周期相同的匀速圆周运动, 现测得两星中心距离为 $\mathrm{R}$, 其运动周期为 $\mathrm{T}$, 若万有引力恒量为 $\mathrm{G}$,则两星的总质量为 ( ) A: $\frac{4 \pi^{2} R^{3}}{G T^{2}}$ B: $\frac{4 \pi^{2} R^{2}}{G T^{2}}$ C: $\frac{4 \pi^{2} R^{2}}{G T^{3}}$ D: $\frac{4 \pi^{2} R^{3}}{G T^{3}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

text-only

Astronomy_1108

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.b. Jupiter has a period of 4332.589 days and Saturn has a period of 10759.22 days (where 1 day $=24$ hours). Note: be careful as your calculations will be very sensitive to rounding errors. iii. Some astronomers have suggested that the 'Star of Bethlehem' seen by the magi ('wise men') on their way to Jesus' birth was in fact a great conjunction. Use your average synodic period to find the date of the great conjunction in the first decade $\mathrm{BC}$ and give your answer to the nearest month. [Note: be careful with $\mathrm{BC}$ years as year 0 in your calculation is equivalent to $1 \mathrm{BC}$, since 31 st December $1 \mathrm{BC}$ is followed by 1 st January $1 \mathrm{AD}$ ]

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: 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: b. Jupiter has a period of 4332.589 days and Saturn has a period of 10759.22 days (where 1 day $=24$ hours). Note: be careful as your calculations will be very sensitive to rounding errors. iii. Some astronomers have suggested that the 'Star of Bethlehem' seen by the magi ('wise men') on their way to Jesus' birth was in fact a great conjunction. Use your average synodic period to find the date of the great conjunction in the first decade $\mathrm{BC}$ and give your answer to the nearest month. [Note: be careful with $\mathrm{BC}$ years as year 0 in your calculation is equivalent to $1 \mathrm{BC}$, since 31 st December $1 \mathrm{BC}$ is followed by 1 st January $1 \mathrm{AD}$ ] 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_818

An astronomer observes that a Solar type star has an apparent V magnitude of 6.73 when seen from the Earth. Assuming that the average interstellar extinction in V is $1.00 \mathrm{mag} / \mathrm{kpc}$, determine the distance between this star and the Solar system. A: $11.5 \mathrm{pc}$ B: $49.5 \mathrm{pc}$ C: $34.2 \mathrm{pc}$ D: $23.7 \mathrm{pc}$ E: $18.9 \mathrm{pc}$

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: An astronomer observes that a Solar type star has an apparent V magnitude of 6.73 when seen from the Earth. Assuming that the average interstellar extinction in V is $1.00 \mathrm{mag} / \mathrm{kpc}$, determine the distance between this star and the Solar system. A: $11.5 \mathrm{pc}$ B: $49.5 \mathrm{pc}$ C: $34.2 \mathrm{pc}$ D: $23.7 \mathrm{pc}$ E: $18.9 \mathrm{pc}$ You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D, E].

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Astronomy

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Astronomy_43

如图所示, 圆轨道与椭圆轨道相切于 $A$ 点, $B$ 点是椭圆轨道的远地点. 人造地球卫星在圆轨道上运行的速率为 $v 1$, 在椭圆轨道上运行经过 $B$ 点时速率为 $v_{2}$, 则 [图1] A: $v_{1}>v_{2}$ B: $v_{1}=v_{2}$ C: $v_{1}<v_{2}$ D: 以上说法都不正确

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, 圆轨道与椭圆轨道相切于 $A$ 点, $B$ 点是椭圆轨道的远地点. 人造地球卫星在圆轨道上运行的速率为 $v 1$, 在椭圆轨道上运行经过 $B$ 点时速率为 $v_{2}$, 则 [图1] A: $v_{1}>v_{2}$ B: $v_{1}=v_{2}$ C: $v_{1}<v_{2}$ D: 以上说法都不正确 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_545

米歇尔$\cdot$麦耶和迪迪埃$\cdot$奎洛兹因为发现了第一颗太阳系外行星一飞马座 $51 b$ 而获得 2019 年诺贝尔物理学奖。如图所示, 飞马座 $51 b$ 与恒星构成双星系统, 绕共同的圆心 $O$做匀速圆周运动, 它们的质量分别为 $m_{1} 、 m_{2}$ 。下列关于飞马座 $51 b$ 与恒星的说法正确的是 ( ) [图1] A: 轨道半径之比为 $m_{1}: m_{2}$ B: 线速度大小之比为 $m_{1}: m_{2}$ C: 加速度大小之比为 $m_{2}: m_{1}$ D: 向心力大小之比为 $m_{2}: m_{1}$

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: 米歇尔$\cdot$麦耶和迪迪埃$\cdot$奎洛兹因为发现了第一颗太阳系外行星一飞马座 $51 b$ 而获得 2019 年诺贝尔物理学奖。如图所示, 飞马座 $51 b$ 与恒星构成双星系统, 绕共同的圆心 $O$做匀速圆周运动, 它们的质量分别为 $m_{1} 、 m_{2}$ 。下列关于飞马座 $51 b$ 与恒星的说法正确的是 ( ) [图1] A: 轨道半径之比为 $m_{1}: m_{2}$ B: 线速度大小之比为 $m_{1}: m_{2}$ C: 加速度大小之比为 $m_{2}: m_{1}$ D: 向心力大小之比为 $m_{2}: m_{1}$ 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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Astronomy

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multi-modal

Astronomy_275

2021 年 10 月 16 日, 神舟十三号载人飞船采用自主快速交会对接模式成功对接于天和核心舱径向端口，对接过程简化如图所示。神舟十三号先到达天和核心舱轨道正下方 $d_{l}=200$ 米的第一停泊点并保持相对静止, 完成各种测控后, 开始沿地心与天和核心舱连线 (径向) 向天和核心舱靠近, 到距离天和核心舱 $d_{2}=19$ 米的第二停泊点短暂驻留,完成各种测控后, 继续径向靠近, 以很小的相对速度完成精准的端口对接。对接技术非常复杂, 故做如下简化。假设地球是半径为 $R_{0}$ 的标准球体，地表重力加速度为 $g$, 忽略自转; 核心舱轨道是半径为 $R$ 的正圆; 神舟十三号质量为 $m_{l}$, 对接前组合体的总质量为 $m_{2}$ ；忽略对接前后神舟十三号质量的变化。 神舟十三号安装有几十台微动力火箭发动机, 用以控制其各种平动和转动, 维持在第一停泊点时, 需要开启某些发动机, 求发动机所提供推力 $F$ 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: 2021 年 10 月 16 日, 神舟十三号载人飞船采用自主快速交会对接模式成功对接于天和核心舱径向端口，对接过程简化如图所示。神舟十三号先到达天和核心舱轨道正下方 $d_{l}=200$ 米的第一停泊点并保持相对静止, 完成各种测控后, 开始沿地心与天和核心舱连线 (径向) 向天和核心舱靠近, 到距离天和核心舱 $d_{2}=19$ 米的第二停泊点短暂驻留,完成各种测控后, 继续径向靠近, 以很小的相对速度完成精准的端口对接。对接技术非常复杂, 故做如下简化。假设地球是半径为 $R_{0}$ 的标准球体，地表重力加速度为 $g$, 忽略自转; 核心舱轨道是半径为 $R$ 的正圆; 神舟十三号质量为 $m_{l}$, 对接前组合体的总质量为 $m_{2}$ ；忽略对接前后神舟十三号质量的变化。 神舟十三号安装有几十台微动力火箭发动机, 用以控制其各种平动和转动, 维持在第一停泊点时, 需要开启某些发动机, 求发动机所提供推力 $F$ 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

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Astronomy

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multi-modal

Astronomy_1200

A 10-inch refracting telescope with focal ratio (defined as the ratio of the focal length and aperture) of 10 is used with a $25 \mathrm{~mm}$ focal length eyepiece. What is the magnifying power of the telescope? $(1$ inch $=2.54 \mathrm{~cm}$ ) A: $10 \mathrm{x}$ B: $50 \mathrm{x}$ C: $100 \mathrm{x}$ D: $200 x$

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 10-inch refracting telescope with focal ratio (defined as the ratio of the focal length and aperture) of 10 is used with a $25 \mathrm{~mm}$ focal length eyepiece. What is the magnifying power of the telescope? $(1$ inch $=2.54 \mathrm{~cm}$ ) A: $10 \mathrm{x}$ B: $50 \mathrm{x}$ C: $100 \mathrm{x}$ D: $200 x$ 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_1008

Given an overwhelmingly large piece of paper, with a thickness of $10 \mu \mathrm{m}$, approximately how many times do you need to fold it in half (theoretically!) for the thickness of the final stack to reach from Earth to the Sun (1 $\mathrm{AU})$ ? A: 40 B: 50 C: 60 D: 70

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: Given an overwhelmingly large piece of paper, with a thickness of $10 \mu \mathrm{m}$, approximately how many times do you need to fold it in half (theoretically!) for the thickness of the final stack to reach from Earth to the Sun (1 $\mathrm{AU})$ ? A: 40 B: 50 C: 60 D: 70 You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

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Astronomy

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Astronomy_1014

The Ring Nebula, M57, is surrounded by thin shells of out-flowing material, the outermost (and faintest) of which is measured to have an angular diameter of 230 arcseconds (where 1 arcsecond $=1 / 3600^{\text {th }}$ of a degree). Given the nebula is $787 \mathrm{pc}$ from Earth, estimate its radius. A: $0.44 \mathrm{pc}$ B: $0.88 \mathrm{pc}$ C: $25 \mathrm{pc}$ D: $50 \mathrm{pc}$

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: The Ring Nebula, M57, is surrounded by thin shells of out-flowing material, the outermost (and faintest) of which is measured to have an angular diameter of 230 arcseconds (where 1 arcsecond $=1 / 3600^{\text {th }}$ of a degree). Given the nebula is $787 \mathrm{pc}$ from Earth, estimate its radius. A: $0.44 \mathrm{pc}$ B: $0.88 \mathrm{pc}$ C: $25 \mathrm{pc}$ D: $50 \mathrm{pc}$ You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

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Astronomy

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Astronomy_235

某行星和地球绕太阳公转的轨道均可视为圆, 每过 $N$ 年, 该行星会运行到日地连线的延长线上，如图所示，下列分析正确的是（） [图1] A: 太阳对该行星的引力小于太阳对地球的引力 B: 行星、地球、太阳三者共线的时间间隔为 $0.25 \mathrm{~N}$ 年 C: 该行星的公转周期为 $\frac{N}{N-1}$ 年 D: 该行星和地球的线速度之比为 $\left(\frac{N-1}{N}\right)^{\frac{1}{3}}$

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: 某行星和地球绕太阳公转的轨道均可视为圆, 每过 $N$ 年, 该行星会运行到日地连线的延长线上，如图所示，下列分析正确的是（） [图1] A: 太阳对该行星的引力小于太阳对地球的引力 B: 行星、地球、太阳三者共线的时间间隔为 $0.25 \mathrm{~N}$ 年 C: 该行星的公转周期为 $\frac{N}{N-1}$ 年 D: 该行星和地球的线速度之比为 $\left(\frac{N-1}{N}\right)^{\frac{1}{3}}$ 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]

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Astronomy

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multi-modal

Astronomy_607

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

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题包含多个待求解的量。 问题: 人造卫星在绕地球运行时, 会遇到稀薄大气的阻力。如果不进行必要的轨道维持,稀薄大气对卫星的这种微小阻力会导致卫星轨道半径逐渐减小, 以至最终落回地球。这个过程是非常漫长的, 因此卫星每一圈的运动仍可以认为是匀速圆周运动。规定两质点相距无穷远时的引力势能为零, 理论上可以得出质量分别 $m_{1} 、 m_{2}$ 的两个物体相距 $r$ 时,系统的引力势能为 $E_{p}=\frac{G m_{1} m_{2}}{r}$ 。已知人造卫星的质量为 $m$, 某时刻绕地球做匀速圆周运动的轨道半径为 $r$, 地球半径为 $R$, 地球表面附近的重力加速度为 $g$ 。 求此时刻卫星的瞬时速度大小 $v$ 和卫星的机械能 $E$ 机。 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你的最终解答的量应该按以下顺序输出：[求此时刻卫星的瞬时速度大小 $v$ , 卫星的机械能 $E$] 它们的答案类型依次是[表达式, 表达式] 你需要在输出的最后用以下格式总结答案：“最终答案是\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5

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Astronomy_835

Which of the following statements is wrong? A: It is believed that elements with atomic number greater than that of iron are formed mostly by the explosion of supernovas. B: What holds a star together is the hydrostatic equilibrium between pressure and gravity. C: The granulations of the Sun happen on its corona. D: Protostars are actually not stars because their main source of heat is not fusion. E: The earlier type the main-sequence star, the more massive it is.

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 statements is wrong? A: It is believed that elements with atomic number greater than that of iron are formed mostly by the explosion of supernovas. B: What holds a star together is the hydrostatic equilibrium between pressure and gravity. C: The granulations of the Sun happen on its corona. D: Protostars are actually not stars because their main source of heat is not fusion. E: The earlier type the main-sequence star, the more massive it is. 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_713

人类设想在赤道平面内建造垂直于地面并延伸到太空的电梯, 又称“太空电梯”如图甲所示。图乙中, 图线 $A$ 表示地球引力对航天员产生的加速度大小与航天员距地心的距离 $r$ 的关系，图线 $B$ 表示航天员相对地面静止时而产生的向心加速度大小与 $r$ 的关系。 图乙中 $R$ (地球半径), $r_{0}$ 为已知量, 地球自转的周期为 $T$, 引力常量为 $G$, 下列说法正确的有 ( ) [图1] 图甲 [图2] 图乙 A: 太空电梯停在 $r_{0}$ 处时, 航天员对电梯舱的弹力为 0 B: 地球的质量为 $\frac{4 \pi^{2} r_{0}^{3}}{G T^{2}}$ C: 地球的第一宇宙速度为 $\frac{2 \pi r_{0}}{T} \sqrt{\frac{G r_{0}}{R}}$ D: 随着 $r$ 的增大, 航天员对电梯舱的弹力逐渐减小

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 人类设想在赤道平面内建造垂直于地面并延伸到太空的电梯, 又称“太空电梯”如图甲所示。图乙中, 图线 $A$ 表示地球引力对航天员产生的加速度大小与航天员距地心的距离 $r$ 的关系，图线 $B$ 表示航天员相对地面静止时而产生的向心加速度大小与 $r$ 的关系。 图乙中 $R$ (地球半径), $r_{0}$ 为已知量, 地球自转的周期为 $T$, 引力常量为 $G$, 下列说法正确的有 ( ) [图1] 图甲 [图2] 图乙 A: 太空电梯停在 $r_{0}$ 处时, 航天员对电梯舱的弹力为 0 B: 地球的质量为 $\frac{4 \pi^{2} r_{0}^{3}}{G T^{2}}$ C: 地球的第一宇宙速度为 $\frac{2 \pi r_{0}}{T} \sqrt{\frac{G r_{0}}{R}}$ D: 随着 $r$ 的增大, 航天员对电梯舱的弹力逐渐减小 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

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multi-modal

Astronomy_294

18 世纪, 数学家莫佩尔蒂, 哲学家伏尔泰曾经设想“穿透”地球 假设能够沿着地球两极连线开丵一条沿着地轴的隧道贯穿地球, 一个人可以从北极入口由静止自由落入隧道中, 忽略一切阻力, 此人可以从南极出口飞出, (已知此人的质量 $m=50 \mathrm{~kg}$; 地球表面处重力加速度 $g$ 取 $10 \mathrm{~m} / \mathrm{s}^{2}$; 地球半径 $R=6.4 \times 10^{6} \mathrm{~m}$; 假设地球可视为质量分布均匀的球体。均匀球壳对壳内任一点的质点合引力为零）则以下说法正确的是（） A: 人与地球构成的系统, 由于重力发生变化, 故机械能不守恒 B: 人在下落过程中, 受到的万有引力与到地心的距离成正比 C: 人从北极开始下落, 到刚好经过地心的过程, 万有引力对人做功 $W=3.2 \times 10^{9} \mathrm{~J}$ D: 当人下落经过距地心 $\frac{1}{2} R$ 瞬间, 人的瞬时速度大小为 $4 \sqrt{3} \times 10^{3} \mathrm{~m} / \mathrm{s}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 18 世纪, 数学家莫佩尔蒂, 哲学家伏尔泰曾经设想“穿透”地球 假设能够沿着地球两极连线开丵一条沿着地轴的隧道贯穿地球, 一个人可以从北极入口由静止自由落入隧道中, 忽略一切阻力, 此人可以从南极出口飞出, (已知此人的质量 $m=50 \mathrm{~kg}$; 地球表面处重力加速度 $g$ 取 $10 \mathrm{~m} / \mathrm{s}^{2}$; 地球半径 $R=6.4 \times 10^{6} \mathrm{~m}$; 假设地球可视为质量分布均匀的球体。均匀球壳对壳内任一点的质点合引力为零）则以下说法正确的是（） A: 人与地球构成的系统, 由于重力发生变化, 故机械能不守恒 B: 人在下落过程中, 受到的万有引力与到地心的距离成正比 C: 人从北极开始下落, 到刚好经过地心的过程, 万有引力对人做功 $W=3.2 \times 10^{9} \mathrm{~J}$ D: 当人下落经过距地心 $\frac{1}{2} R$ 瞬间, 人的瞬时速度大小为 $4 \sqrt{3} \times 10^{3} \mathrm{~m} / \mathrm{s}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

text-only

Astronomy_1112

In order to move a spacecraft between orbits we must apply a thrust using rockets, which changes the velocity of the spacecraft by $\Delta v$. In this question we will ignore changes in the mass of the spacecraft due to the burning of fuel. For an object of mass $m$ in a circular orbit of radius $r$ around an object with mass $M$ (where $m \ll M$ ) the orbital velocity, $v_{\text {orb }}$, is given by the formula $v_{\text {orb }}=\sqrt{\frac{G M}{r}}$.c. Approximating Mars' orbit as circular with a radius of $1.52 \mathrm{AU}$, calculate the $\triangle v$ to go from Earth LEO to Mars i.e. $\Delta v=\left|\Delta v_{A}\right|+\left|\Delta v_{B}\right|$. Compare your answer to the $\Delta v$ to reach Earth LEO.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: In order to move a spacecraft between orbits we must apply a thrust using rockets, which changes the velocity of the spacecraft by $\Delta v$. In this question we will ignore changes in the mass of the spacecraft due to the burning of fuel. For an object of mass $m$ in a circular orbit of radius $r$ around an object with mass $M$ (where $m \ll M$ ) the orbital velocity, $v_{\text {orb }}$, is given by the formula $v_{\text {orb }}=\sqrt{\frac{G M}{r}}$. problem: c. Approximating Mars' orbit as circular with a radius of $1.52 \mathrm{AU}$, calculate the $\triangle v$ to go from Earth LEO to Mars i.e. $\Delta v=\left|\Delta v_{A}\right|+\left|\Delta v_{B}\right|$. Compare your answer to the $\Delta v$ to reach Earth LEO. 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

text-only

Astronomy_749

As the life of a star progresses, heavy elements are produced. The elements form layers around the star in this order (starting from the outer layer): A: $\mathrm{H} \rightarrow \mathrm{He} \rightarrow \mathrm{Li} \rightarrow \mathrm{N} \rightarrow \mathrm{O} \rightarrow \mathrm{Si} \rightarrow \mathrm{Fe}$ B: $\mathrm{H} \rightarrow \mathrm{He} \rightarrow \mathrm{C} \rightarrow \mathrm{O} \rightarrow \mathrm{Ne} \rightarrow \mathrm{Si} \rightarrow \mathrm{Fe}$ C: $\mathrm{H} \rightarrow \mathrm{He} \rightarrow \mathrm{Li} \rightarrow \mathrm{O} \rightarrow \mathrm{Ne} \rightarrow \mathrm{Si} \rightarrow \mathrm{Fe}$ D: $\mathrm{H} \rightarrow \mathrm{He} \rightarrow \mathrm{C} \rightarrow \mathrm{N} \rightarrow \mathrm{O} \rightarrow \mathrm{Si} \rightarrow \mathrm{Fe}$

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: As the life of a star progresses, heavy elements are produced. The elements form layers around the star in this order (starting from the outer layer): A: $\mathrm{H} \rightarrow \mathrm{He} \rightarrow \mathrm{Li} \rightarrow \mathrm{N} \rightarrow \mathrm{O} \rightarrow \mathrm{Si} \rightarrow \mathrm{Fe}$ B: $\mathrm{H} \rightarrow \mathrm{He} \rightarrow \mathrm{C} \rightarrow \mathrm{O} \rightarrow \mathrm{Ne} \rightarrow \mathrm{Si} \rightarrow \mathrm{Fe}$ C: $\mathrm{H} \rightarrow \mathrm{He} \rightarrow \mathrm{Li} \rightarrow \mathrm{O} \rightarrow \mathrm{Ne} \rightarrow \mathrm{Si} \rightarrow \mathrm{Fe}$ D: $\mathrm{H} \rightarrow \mathrm{He} \rightarrow \mathrm{C} \rightarrow \mathrm{N} \rightarrow \mathrm{O} \rightarrow \mathrm{Si} \rightarrow \mathrm{Fe}$ 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_941

The Gaia spacecraft, launched in 2013, is designed to map the sky with unprecedented precision. Its third data release (from June 2022) had $1.8 \times 10^{9}$ objects detected of which $1.9 \times 10^{6}$ were highly likely to be quasars (distant galaxies with an actively feeding black hole responsible for their light output). Assuming the quasars are evenly distributed across the whole sky, what is the approximate average angular separation between them? A: $\sim 0.001^{\circ}$ B: $\sim 0.01^{\circ}$ C: $\sim 0.1^{\circ}$ D: $\sim 1.0^{\circ}$

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: The Gaia spacecraft, launched in 2013, is designed to map the sky with unprecedented precision. Its third data release (from June 2022) had $1.8 \times 10^{9}$ objects detected of which $1.9 \times 10^{6}$ were highly likely to be quasars (distant galaxies with an actively feeding black hole responsible for their light output). Assuming the quasars are evenly distributed across the whole sky, what is the approximate average angular separation between them? A: $\sim 0.001^{\circ}$ B: $\sim 0.01^{\circ}$ C: $\sim 0.1^{\circ}$ D: $\sim 1.0^{\circ}$ You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

SC

Astronomy

EN

text-only

Astronomy_426

如图所示, $\mathrm{A}$ 是地球同步卫星, 另一个卫星 $\mathrm{B}$ 的圆轨道位于赤道平面内, 卫星 B 的运行周期为 $T$ 。已知地球半径为 $R$, 地球自转周期为 $T_{0}$, 地球表面的重力加速度为 $g$ 求卫星 B 距离地面高度; [图1]

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 如图所示, $\mathrm{A}$ 是地球同步卫星, 另一个卫星 $\mathrm{B}$ 的圆轨道位于赤道平面内, 卫星 B 的运行周期为 $T$ 。已知地球半径为 $R$, 地球自转周期为 $T_{0}$, 地球表面的重力加速度为 $g$ 求卫星 B 距离地面高度; [图1] 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

EX

Astronomy

ZH

multi-modal

Astronomy_996

In the video game Halo, players are able to move around on the inner edge of a ring structure, which has a radius of $5000 \mathrm{~km}$. Roughly how many rotations per (Earth) day would the ring need to achieve in order for the players' characters to experience Earth-like gravity whilst walking around? A: 15 B: 17 C: 19 D: 21

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: In the video game Halo, players are able to move around on the inner edge of a ring structure, which has a radius of $5000 \mathrm{~km}$. Roughly how many rotations per (Earth) day would the ring need to achieve in order for the players' characters to experience Earth-like gravity whilst walking around? A: 15 B: 17 C: 19 D: 21 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_1120

All stars lose mass during their lifetimes due to two main routes: particles escaping their surface (referred to as the stellar wind), and the mass defect of the nuclear reactions occurring in their cores. In practice, the mass loss rate can vary quite considerably during a star's lifetime, particularly once it has left the main sequence when the stellar wind can become much more substantial. Wolf-Rayet stars are massive stars near the end of their lives, presumed to be the in the stage just before a supernova, and are losing substantial amounts of mass due to very fast stellar winds. This deposits considerable energy into the surrounding interstellar medium (ISM) and can sweep up material into a thin bubble around the star, visible as a type of planetary nebula. [figure1] Figure 1: The nebula NGC 2359 around the Wolf-Rayet star WR7. The nebula is known as Thor's Helmet due to its resemblance to the helmet worn by the character from the Marvel Comics series. Credit: Star Shadows Remote Observatory and PROMPT/UNC.c. The Wolf-Rayet star WR7 is in the constellation of Canis Major and its strong winds are responsible for the nebula known as Thor's Helmet. The star has a mass of $16 M_{\odot}$, a radius of $1.41 R_{\odot}$ and a surface temperature of $112000 \mathrm{~K}$, with a measured $v_{\infty}$ of $1545 \mathrm{~km} \mathrm{~s}^{-1}$. i. Predict the mass loss rate of WR7 based upon the properties of the star, assuming the single scattering limit. Give your answer in units of $M_{\odot} \mathrm{yr}^{-1}$.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is an expression. Here is some context information for this question, which might assist you in solving it: All stars lose mass during their lifetimes due to two main routes: particles escaping their surface (referred to as the stellar wind), and the mass defect of the nuclear reactions occurring in their cores. In practice, the mass loss rate can vary quite considerably during a star's lifetime, particularly once it has left the main sequence when the stellar wind can become much more substantial. Wolf-Rayet stars are massive stars near the end of their lives, presumed to be the in the stage just before a supernova, and are losing substantial amounts of mass due to very fast stellar winds. This deposits considerable energy into the surrounding interstellar medium (ISM) and can sweep up material into a thin bubble around the star, visible as a type of planetary nebula. [figure1] Figure 1: The nebula NGC 2359 around the Wolf-Rayet star WR7. The nebula is known as Thor's Helmet due to its resemblance to the helmet worn by the character from the Marvel Comics series. Credit: Star Shadows Remote Observatory and PROMPT/UNC. problem: c. The Wolf-Rayet star WR7 is in the constellation of Canis Major and its strong winds are responsible for the nebula known as Thor's Helmet. The star has a mass of $16 M_{\odot}$, a radius of $1.41 R_{\odot}$ and a surface temperature of $112000 \mathrm{~K}$, with a measured $v_{\infty}$ of $1545 \mathrm{~km} \mathrm{~s}^{-1}$. i. Predict the mass loss rate of WR7 based upon the properties of the star, assuming the single scattering limit. Give your answer in units of $M_{\odot} \mathrm{yr}^{-1}$. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is an expression without equals signs, e.g. ANSWER=\frac{1}{2} g t^2

EX

Astronomy

EN

multi-modal

Astronomy_1167

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 $q=0.251 \rho X^{2} T_{6}^{-2 / 3} e^{-33.80 T_{6}^{-1 / 3}} \mathrm{~W} \mathrm{~kg}^{-1}$.b. The Sun is composed predominantly of ionized hydrogen and helium, with approximate mass fractions $X=0.70$ and $Y=0.30$ respectively (taken to be constant throughout the Sun). i. Show that the kinetic energy per unit mass of the solar plasma is the formula given.

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: b. The Sun is composed predominantly of ionized hydrogen and helium, with approximate mass fractions $X=0.70$ and $Y=0.30$ respectively (taken to be constant throughout the Sun). i. Show that the kinetic energy per unit mass of the solar plasma is the formula given. 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_243

卫星 $\mathrm{A} 、 \mathrm{~B}$ 的运行方向相同, 其中 $\mathrm{B}$ 为近地卫星. 某时刻, 两卫星相距最近 $(\mathrm{O} 、 \mathrm{~B} 、$ $\mathrm{A}$ 在同一直线上). 已知地球半径为 $R$, 卫星 $\mathrm{A}$ 离地心 $\mathrm{O}$ 的距离是卫星 $\mathrm{B}$ 离地心 $\mathrm{O}$ 距离的 4 倍, 地球表面的重力加速度为 $g$, 则( ) [图1] A: 卫星 $\mathrm{A} 、 \mathrm{~B}$ 运行的加速度大小之比 $\frac{a_{A}}{a_{B}}=\frac{1}{4}$ B: 卫星 $\mathrm{A} 、 \mathrm{~B}$ 运行的周期之比 $\frac{T_{A}}{T_{B}}=\frac{4}{1}$ C: 卫星 $\mathrm{A} 、 \mathrm{~B}$ 运行的线速度大小之比 $\frac{v_{A}}{v_{B}}=\frac{1}{2}$ D: 卫星 $\mathrm{A} 、 \mathrm{~B}$ 至少要经过时间 $t=\frac{16 \pi}{7} \sqrt{\frac{R}{g}}$, 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。 这是一个多选题（有多个正确答案）。 问题: 卫星 $\mathrm{A} 、 \mathrm{~B}$ 的运行方向相同, 其中 $\mathrm{B}$ 为近地卫星. 某时刻, 两卫星相距最近 $(\mathrm{O} 、 \mathrm{~B} 、$ $\mathrm{A}$ 在同一直线上). 已知地球半径为 $R$, 卫星 $\mathrm{A}$ 离地心 $\mathrm{O}$ 的距离是卫星 $\mathrm{B}$ 离地心 $\mathrm{O}$ 距离的 4 倍, 地球表面的重力加速度为 $g$, 则( ) [图1] A: 卫星 $\mathrm{A} 、 \mathrm{~B}$ 运行的加速度大小之比 $\frac{a_{A}}{a_{B}}=\frac{1}{4}$ B: 卫星 $\mathrm{A} 、 \mathrm{~B}$ 运行的周期之比 $\frac{T_{A}}{T_{B}}=\frac{4}{1}$ C: 卫星 $\mathrm{A} 、 \mathrm{~B}$ 运行的线速度大小之比 $\frac{v_{A}}{v_{B}}=\frac{1}{2}$ D: 卫星 $\mathrm{A} 、 \mathrm{~B}$ 至少要经过时间 $t=\frac{16 \pi}{7} \sqrt{\frac{R}{g}}$, 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应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_1187

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 $$ \dot{f}_{\mathrm{GW}}=\frac{96}{5} \pi^{8 / 3}\left(\frac{G \mathcal{M}}{c^{3}}\right)^{5 / 3} f_{\mathrm{GW}}^{11 / 3} $$b. For NGC 4993 we measure $R_{e}=15.5$ arcseconds, $\sigma=171 \mathrm{~km} \mathrm{~s}^{-1}$, and $\left\langle 1_{r}>_{e}=407 \mathrm{~L}_{\odot} \mathrm{pc}^{-2}\right.$. Given that the scatter in the FP relation introduces an uncertainty in D of $\pm 17 \%$, calculate the distance to the galaxy (in Mpc) and its (absolute) uncertainty using the FP relation.

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: b. For NGC 4993 we measure $R_{e}=15.5$ arcseconds, $\sigma=171 \mathrm{~km} \mathrm{~s}^{-1}$, and $\left\langle 1_{r}>_{e}=407 \mathrm{~L}_{\odot} \mathrm{pc}^{-2}\right.$. Given that the scatter in the FP relation introduces an uncertainty in D of $\pm 17 \%$, calculate the distance to the galaxy (in Mpc) and its (absolute) uncertainty using the FP relation. 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{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

text-only

Astronomy_703

天文学家通过观测两个黑洞并合的事件, 间接验证了引力波的存在。该事件中甲、乙两个黑洞的质量分别为太阳质量的 36 倍和 29 倍, 假设这两个黑洞绕它们连线上的某 点做圆周运动, 且两个黑洞的间距缓慢减小。若该双星系统在运动过程中, 各自质量不变且不受其他星系的影响，则关于这两个黑洞的运动，下列说法正确的是（） A: 甲、乙两个黑洞运行的线速度大小之比为 $36: 29$ B: 甲、乙两个黑洞运行的角速度大小始终相等 C: 随着甲、乙两个黑洞的间距缓慢减小，它们运行的周期也在减小 D: 甲、乙两个黑洞做圆周运动的向心加速度大小始终相等

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 天文学家通过观测两个黑洞并合的事件, 间接验证了引力波的存在。该事件中甲、乙两个黑洞的质量分别为太阳质量的 36 倍和 29 倍, 假设这两个黑洞绕它们连线上的某 点做圆周运动, 且两个黑洞的间距缓慢减小。若该双星系统在运动过程中, 各自质量不变且不受其他星系的影响，则关于这两个黑洞的运动，下列说法正确的是（） A: 甲、乙两个黑洞运行的线速度大小之比为 $36: 29$ B: 甲、乙两个黑洞运行的角速度大小始终相等 C: 随着甲、乙两个黑洞的间距缓慢减小，它们运行的周期也在减小 D: 甲、乙两个黑洞做圆周运动的向心加速度大小始终相等 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应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

text-only

Astronomy_1029

In the early 1900s, Henrietta Leavitt made several observations of variable stars from the Harvard College Observatory that led her to propose that the period of variation was related to the intrinsic brightness of the star. In particular, for a class of variable stars known as Cepheids, there was a strong power law relating luminosity to period, and hence a straight line on a log-log graph. Since magnitudes are $\propto \log L$, this can be described by the empirical relation $$ \langle\mathcal{M}\rangle=-2.43(\log P-1)-4.05 $$ where $P$ is the period measured in days and $\langle\mathcal{M}\rangle$ is the mean absolute magnitude of the star, defined as the apparent magnitude measured from a distance of $10 \mathrm{pc}$. The apparent and absolute magnitudes are related as $$ m-\mathfrak{M}=5 \log d-5 $$ where $d$ is the distance measured in pc. [figure1] Figure 1: Left: Colour composite view of the circumstellar nebula around RS Puppis assembled from HST images. Credit: NASA, ESA, and the Hubble Heritage Team (STScI/AURA)-Hubble/Europe Collaboration. Right: The light curve of RS Puppis. Phase 0 corresponds to the when the star is at its brightest, and the data has been folded over a known period of 41.5 days. Credit: AAVSO / P. Kervella et al. (2017). In the past, distances to stars could only be determined using something called the parallax method, where the angular shift of a star relative to the background stars was measured at two points in the Earth's orbit six months apart. The parallax angle is the same as the angle subtended by 1 au at the distance of the star. In Henrietta's time, parallax measurements only reached about $100 \mathrm{pc}$, however her method with Cepheid variables extended this range to more than $10^{6} \mathrm{pc}$, enabling much better determination of distances across the Milky Way and into nearby galaxies. The variable star RS Puppis (see Figure 1) is one of the biggest and brightest known Cepheids in the Milky Way galaxy and has one of the longest periods for this class of star at 41.5 days. Measure the mean apparent magnitude from Figure 1, and find the distance to the star, given that absorption of the light by interstellar dust means the star appears 1.42 magnitudes fainter.

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. problem: In the early 1900s, Henrietta Leavitt made several observations of variable stars from the Harvard College Observatory that led her to propose that the period of variation was related to the intrinsic brightness of the star. In particular, for a class of variable stars known as Cepheids, there was a strong power law relating luminosity to period, and hence a straight line on a log-log graph. Since magnitudes are $\propto \log L$, this can be described by the empirical relation $$ \langle\mathcal{M}\rangle=-2.43(\log P-1)-4.05 $$ where $P$ is the period measured in days and $\langle\mathcal{M}\rangle$ is the mean absolute magnitude of the star, defined as the apparent magnitude measured from a distance of $10 \mathrm{pc}$. The apparent and absolute magnitudes are related as $$ m-\mathfrak{M}=5 \log d-5 $$ where $d$ is the distance measured in pc. [figure1] Figure 1: Left: Colour composite view of the circumstellar nebula around RS Puppis assembled from HST images. Credit: NASA, ESA, and the Hubble Heritage Team (STScI/AURA)-Hubble/Europe Collaboration. Right: The light curve of RS Puppis. Phase 0 corresponds to the when the star is at its brightest, and the data has been folded over a known period of 41.5 days. Credit: AAVSO / P. Kervella et al. (2017). In the past, distances to stars could only be determined using something called the parallax method, where the angular shift of a star relative to the background stars was measured at two points in the Earth's orbit six months apart. The parallax angle is the same as the angle subtended by 1 au at the distance of the star. In Henrietta's time, parallax measurements only reached about $100 \mathrm{pc}$, however her method with Cepheid variables extended this range to more than $10^{6} \mathrm{pc}$, enabling much better determination of distances across the Milky Way and into nearby galaxies. The variable star RS Puppis (see Figure 1) is one of the biggest and brightest known Cepheids in the Milky Way galaxy and has one of the longest periods for this class of star at 41.5 days. Measure the mean apparent magnitude from Figure 1, and find the distance to the star, given that absorption of the light by interstellar dust means the star appears 1.42 magnitudes fainter. 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 pc, 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_1013

The cubesat CAPSTONE will enter a near-rectilinear halo orbit (NRHO) of the Moon on $13^{\text {th }}$ November 2022 as a direct test of the orbit planned for the Lunar Gateway - a space station due to be built in orbit around the Moon by the end of the decade. It is a polar orbit, going from $1500 \mathrm{~km}$ above the North pole to $70000 \mathrm{~km}$ above the South pole. Treating it as an ellipse, what is the eccentricity of the orbit? The Moon has a radius of $1740 \mathrm{~km}$. A: 0.83 B: 0.88 C: 0.91 D: 0.96

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 cubesat CAPSTONE will enter a near-rectilinear halo orbit (NRHO) of the Moon on $13^{\text {th }}$ November 2022 as a direct test of the orbit planned for the Lunar Gateway - a space station due to be built in orbit around the Moon by the end of the decade. It is a polar orbit, going from $1500 \mathrm{~km}$ above the North pole to $70000 \mathrm{~km}$ above the South pole. Treating it as an ellipse, what is the eccentricity of the orbit? The Moon has a radius of $1740 \mathrm{~km}$. A: 0.83 B: 0.88 C: 0.91 D: 0.96 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_158

两颗相距较远的行星 $\mathrm{A} 、 \mathrm{~B}$ 的半径分别为 $R_{A} 、 R_{B}$, 距 $\mathrm{A} 、 \mathrm{~B}$ 行星中心 $r$ 处, 各有一卫星分别围绕行星做匀速圆周运动, 线速度的平方 $v^{2}$ 随半径 $r$ 变化的关系如图甲所示,两图线左端的纵坐标相同; 卫星做匀速圆周运动的周期为 $T, \lg T-\lg r$ 的图像如图乙所示的两平行直线, 它们的截距分别为 $b_{A} 、 b_{B}$. 已知两图像数据均采用国际单位, $b_{\mathrm{B}}-b_{\mathrm{A}}=\lg \sqrt{3}$, 行星可看作质量分布均匀的球体, 忽略行星的自转和其他星球的影响, 下列说法正确的是（ ） [图1] 甲 [图2] A: 图乙中两条直线的斜率均为 $\frac{3}{2}$ B: 行星 A、B 的质量之比为 $1: 3$ C: 行星 A、B 的密度之比为 $1: 9$ D: 行星 A、B 表面的重力加速度大小之比为 $3: 1$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 两颗相距较远的行星 $\mathrm{A} 、 \mathrm{~B}$ 的半径分别为 $R_{A} 、 R_{B}$, 距 $\mathrm{A} 、 \mathrm{~B}$ 行星中心 $r$ 处, 各有一卫星分别围绕行星做匀速圆周运动, 线速度的平方 $v^{2}$ 随半径 $r$ 变化的关系如图甲所示,两图线左端的纵坐标相同; 卫星做匀速圆周运动的周期为 $T, \lg T-\lg r$ 的图像如图乙所示的两平行直线, 它们的截距分别为 $b_{A} 、 b_{B}$. 已知两图像数据均采用国际单位, $b_{\mathrm{B}}-b_{\mathrm{A}}=\lg \sqrt{3}$, 行星可看作质量分布均匀的球体, 忽略行星的自转和其他星球的影响, 下列说法正确的是（ ） [图1] 甲 [图2] A: 图乙中两条直线的斜率均为 $\frac{3}{2}$ B: 行星 A、B 的质量之比为 $1: 3$ C: 行星 A、B 的密度之比为 $1: 9$ D: 行星 A、B 表面的重力加速度大小之比为 $3: 1$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_108

“天问一号”从地球发射后, 在如图甲所示的 $P$ 点沿地火转移轨道到 $Q$ 点, 再依次进入如图乙所示在近火点相切的调相轨道和停泊轨道, 则天问一号 ( ) [图1] 甲 [图2] 乙 A: 从 $P$ 点转移到 $Q$ 点的时间小于 6 个月 B: 在近火点实施制动可以实现从调相轨道转移到停泊轨道 C: 在停泊轨道运行的周期比在调相轨道上运行的周期大 D: 在相等时间内与火星球心的连线在停泊轨道上扫过的面积小于在调相轨道上扫过的面积

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: “天问一号”从地球发射后, 在如图甲所示的 $P$ 点沿地火转移轨道到 $Q$ 点, 再依次进入如图乙所示在近火点相切的调相轨道和停泊轨道, 则天问一号 ( ) [图1] 甲 [图2] 乙 A: 从 $P$ 点转移到 $Q$ 点的时间小于 6 个月 B: 在近火点实施制动可以实现从调相轨道转移到停泊轨道 C: 在停泊轨道运行的周期比在调相轨道上运行的周期大 D: 在相等时间内与火星球心的连线在停泊轨道上扫过的面积小于在调相轨道上扫过的面积 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_595

如图为天文学家观测到的旋浴星系的旋转曲线, 该曲线在旋浴星系发光区之外并没有按天文学家预想的那样, 而是和预想曲线发生了较大偏差, 这引起了科学家们极大的兴趣, 我们知道, 根据人造卫星运行的速度和高度, 就可以测出地球的总质量, 根据地球绕太阳运行的速度和地球与太阳的距离, 就可以测出太阳的总质量, 同理, 根据某个星系内恒星或气团围绕该星系中心运行的速度和它们与“星系中心”的距离, 天文学家就可以估算出这个星系在该恒星或气团所处范围内物质的质量和分布, 经天文学家计算分析得出的结论是: 旋浴星系的总质量远大于星系中可见星体质量的总和, 请根据本题所引用的科普材料判断以下说法正确的是( ) [图1] A: 根据牛顿定律和万有引力定律推导出的旋浴预想曲线应该是图中上面那条曲线 B: 旋浴星系的观测曲线和预想曲线的较大差别说明万有引力定律错了, 需要创建新的定律 C: 旋浴星系的旋转曲线中下降的曲线部分意味着星系中很可能包含了更多的不可见的物质 D: 旋浴星系的旋转曲线中平坦的曲线部分意味着星系中很可能包含了更多的不可见的物质

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: 如图为天文学家观测到的旋浴星系的旋转曲线, 该曲线在旋浴星系发光区之外并没有按天文学家预想的那样, 而是和预想曲线发生了较大偏差, 这引起了科学家们极大的兴趣, 我们知道, 根据人造卫星运行的速度和高度, 就可以测出地球的总质量, 根据地球绕太阳运行的速度和地球与太阳的距离, 就可以测出太阳的总质量, 同理, 根据某个星系内恒星或气团围绕该星系中心运行的速度和它们与“星系中心”的距离, 天文学家就可以估算出这个星系在该恒星或气团所处范围内物质的质量和分布, 经天文学家计算分析得出的结论是: 旋浴星系的总质量远大于星系中可见星体质量的总和, 请根据本题所引用的科普材料判断以下说法正确的是( ) [图1] A: 根据牛顿定律和万有引力定律推导出的旋浴预想曲线应该是图中上面那条曲线 B: 旋浴星系的观测曲线和预想曲线的较大差别说明万有引力定律错了, 需要创建新的定律 C: 旋浴星系的旋转曲线中下降的曲线部分意味着星系中很可能包含了更多的不可见的物质 D: 旋浴星系的旋转曲线中平坦的曲线部分意味着星系中很可能包含了更多的不可见的物质 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_670

如图所示, 甲、乙两颗卫星绕地球做圆周运动, 已知甲卫星的周期为 $N$ 小时, 每过 $9 N$ 小时, 乙卫星都要运动到与甲卫星同居于地球一侧且三者共线的位置上, 则甲、乙两颗卫星的线速度之比为 ( ) [图1] A: $\frac{\sqrt[3]{9}}{2}$ B: $\frac{\sqrt[3]{3}}{2}$ C: $\frac{2}{\sqrt[3]{3}}$ D: $\frac{2}{\sqrt[3]{9}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 如图所示, 甲、乙两颗卫星绕地球做圆周运动, 已知甲卫星的周期为 $N$ 小时, 每过 $9 N$ 小时, 乙卫星都要运动到与甲卫星同居于地球一侧且三者共线的位置上, 则甲、乙两颗卫星的线速度之比为 ( ) [图1] A: $\frac{\sqrt[3]{9}}{2}$ B: $\frac{\sqrt[3]{3}}{2}$ C: $\frac{2}{\sqrt[3]{3}}$ D: $\frac{2}{\sqrt[3]{9}}$ 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_547

如图所示, 由 $\mathrm{A} 、 \mathrm{~B}$ 组成的双星系统, 绕它们连线上的一点做匀速圆周运动, 其运行周期为 $T, \mathrm{~A} 、 \mathrm{~B}$ 间的距离为 $L$, 它们的线速度之比 $\frac{v_{1}}{v_{2}}=2$, 则 ( ) [图1] A: $\mathrm{AB}$ 角速度比为: $\frac{\omega_{A}}{\omega_{B}}=\frac{2}{1}$ B: $\mathrm{AB}$ 质量比为: $\frac{M_{A}}{M_{B}}=\frac{2}{1}$ C: A 星球质量为: $M_{A}=\frac{4 \pi^{2} L^{3}}{G T^{2}}$ D: 两星球质量之和为: $M_{A}+M_{B}=\frac{4 \pi^{2} L^{3}}{G T^{2}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 如图所示, 由 $\mathrm{A} 、 \mathrm{~B}$ 组成的双星系统, 绕它们连线上的一点做匀速圆周运动, 其运行周期为 $T, \mathrm{~A} 、 \mathrm{~B}$ 间的距离为 $L$, 它们的线速度之比 $\frac{v_{1}}{v_{2}}=2$, 则 ( ) [图1] A: $\mathrm{AB}$ 角速度比为: $\frac{\omega_{A}}{\omega_{B}}=\frac{2}{1}$ B: $\mathrm{AB}$ 质量比为: $\frac{M_{A}}{M_{B}}=\frac{2}{1}$ C: A 星球质量为: $M_{A}=\frac{4 \pi^{2} L^{3}}{G T^{2}}$ D: 两星球质量之和为: $M_{A}+M_{B}=\frac{4 \pi^{2} L^{3}}{G T^{2}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

multi-modal

Astronomy_760

If the Earth had the density of a neutron star, what would be the diameter of the Earth? A: between 1 - $100 \mathrm{~m}$ B: between $100-500 \mathrm{~m}$ C: between 500 - $1000 \mathrm{~m}$ D: between $1000-5000 \mathrm{~m}$

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: If the Earth had the density of a neutron star, what would be the diameter of the Earth? A: between 1 - $100 \mathrm{~m}$ B: between $100-500 \mathrm{~m}$ C: between 500 - $1000 \mathrm{~m}$ D: between $1000-5000 \mathrm{~m}$ You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

SC

Astronomy

EN

text-only

Astronomy_1173

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. ii. We can get a good estimate of the central pressure if we use $m$ as our independent variable rather than $r$. Derive an expression for $\mathrm{dm} / \mathrm{dr}$ in terms of $r$ and $\rho$, and hence express $\mathrm{dp} / \mathrm{dm}$ in terms of $m$ and $r$.

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. ii. We can get a good estimate of the central pressure if we use $m$ as our independent variable rather than $r$. Derive an expression for $\mathrm{dm} / \mathrm{dr}$ in terms of $r$ and $\rho$, and hence express $\mathrm{dp} / \mathrm{dm}$ in terms of $m$ and $r$. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is an expression without equals signs, e.g. ANSWER=\frac{1}{2} g t^2

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Astronomy

EN

multi-modal

Astronomy_736

英国《自然》杂志、美国太空网 2017 年 4 月 19 日共同发布消息称，一颗温度适中的岩态行星 LHS 1140b 在经过小型 LHS 1140 矮恒星时发生凌星现象。这颗新发现的超级地球”与恒星的距离、岩石构成以及存在液态水的可能性, 使其成为目前寻找外星生命的最佳选择。假设行星 LHS $1140 b$ 绕 LHS 1140 恒星和地球绕太阳的运动均看作匀速圆周运动, 下表是网上公布的相关数据, 则下列说法正确的是（） | 恒星 | 太阳 | 质量为 $M$ | | :--- | :--- | :--- | | | LHS 1140 | 质量为 $0.6 M$ | | :---: | :---: | :---: | | 行星 | 地球 | 质量为 $m$ <br> 轨道半径为 $r$ | | | LHS 1140b | 质量为 $6.6 m$ <br> 轨道半径为 $1.4 r$ | A: LHS $1140 b$ 与地球运行的速度大小之比为 $\sqrt{\frac{5}{7}}$ B: LHS $1140 b$ 与地球运行的周期之比为 $\frac{7 \sqrt{21}}{15}$ C: LHS $1140 b$ 的第一宇宙速度与地球的第一宇宙速度之比为 $\sqrt{\frac{3}{7}}$ D: LHS $1140 b$ 的密度是地球密度的 $\frac{6.6}{1.4^{3}}$ 倍

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 英国《自然》杂志、美国太空网 2017 年 4 月 19 日共同发布消息称，一颗温度适中的岩态行星 LHS 1140b 在经过小型 LHS 1140 矮恒星时发生凌星现象。这颗新发现的超级地球”与恒星的距离、岩石构成以及存在液态水的可能性, 使其成为目前寻找外星生命的最佳选择。假设行星 LHS $1140 b$ 绕 LHS 1140 恒星和地球绕太阳的运动均看作匀速圆周运动, 下表是网上公布的相关数据, 则下列说法正确的是（） | 恒星 | 太阳 | 质量为 $M$ | | :--- | :--- | :--- | | | LHS 1140 | 质量为 $0.6 M$ | | :---: | :---: | :---: | | 行星 | 地球 | 质量为 $m$ <br> 轨道半径为 $r$ | | | LHS 1140b | 质量为 $6.6 m$ <br> 轨道半径为 $1.4 r$ | A: LHS $1140 b$ 与地球运行的速度大小之比为 $\sqrt{\frac{5}{7}}$ B: LHS $1140 b$ 与地球运行的周期之比为 $\frac{7 \sqrt{21}}{15}$ C: LHS $1140 b$ 的第一宇宙速度与地球的第一宇宙速度之比为 $\sqrt{\frac{3}{7}}$ D: LHS $1140 b$ 的密度是地球密度的 $\frac{6.6}{1.4^{3}}$ 倍 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

ZH

text-only

Astronomy_500

关于黑洞和暗物质(暗物质被称为“世纪之谜” . 它“霸占”了宇宙 $95 \%$ 的地盘, 却摸不到看不着)的问题, 以下说法正确的是(黑洞临界半径公式取为 $c=\sqrt{\frac{2 G M}{r}}, c$ 为光速, $G$ 为万有引力常量, $M$ 为黑洞质量 A: 如果地球成为黑洞的话, 那么它的临界半径为 $r=\frac{v^{2}}{c^{2}} R(R$ 为地球的半径, $v$ 为第二宇宙速度) B: 如果太阳成为黑洞, 那么灿烂的阳光依然存在, 只是太阳光到地球的时间变得更长 C: 有两颗星球(质量分别为 $M_{1}$ 和 $M_{2}$ )的距离为 $L$, 不考虑周围其他星球的影响, 由牛顿运动定律计算所得的周期为 $T$, 由于宇宙充满均匀的暗物质, 所以观察测量所得的周期比 $T$ 大 D: 有两颗星球甲和乙(质量分别为 $M_{1}$ 和 $M_{2}$ ) 的距离为 $L$, 不考虑周围其他星球的影响, 它们运动的周期为 $T$, 如果其中甲的质量减小 $\Delta m$ 而乙的质量增大 $\Delta m$, 距离 $L$不变, 那么它们的周期依然为 $T$

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: 关于黑洞和暗物质(暗物质被称为“世纪之谜” . 它“霸占”了宇宙 $95 \%$ 的地盘, 却摸不到看不着)的问题, 以下说法正确的是(黑洞临界半径公式取为 $c=\sqrt{\frac{2 G M}{r}}, c$ 为光速, $G$ 为万有引力常量, $M$ 为黑洞质量 A: 如果地球成为黑洞的话, 那么它的临界半径为 $r=\frac{v^{2}}{c^{2}} R(R$ 为地球的半径, $v$ 为第二宇宙速度) B: 如果太阳成为黑洞, 那么灿烂的阳光依然存在, 只是太阳光到地球的时间变得更长 C: 有两颗星球(质量分别为 $M_{1}$ 和 $M_{2}$ )的距离为 $L$, 不考虑周围其他星球的影响, 由牛顿运动定律计算所得的周期为 $T$, 由于宇宙充满均匀的暗物质, 所以观察测量所得的周期比 $T$ 大 D: 有两颗星球甲和乙(质量分别为 $M_{1}$ 和 $M_{2}$ ) 的距离为 $L$, 不考虑周围其他星球的影响, 它们运动的周期为 $T$, 如果其中甲的质量减小 $\Delta m$ 而乙的质量增大 $\Delta m$, 距离 $L$不变, 那么它们的周期依然为 $T$ 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_1036

In the early 1900s, Henrietta Leavitt made several observations of variable stars from the Harvard College Observatory that led her to propose that the period of variation was related to the intrinsic brightness of the star. In particular, for a class of variable stars known as Cepheids, there was a strong power law relating luminosity to period, and hence a straight line on a log-log graph. Since magnitudes are $\propto \log L$, this can be described by the empirical relation $$ \langle\mathcal{M}\rangle=-2.43(\log P-1)-4.05 $$ where $P$ is the period measured in days and $\langle\mathcal{M}\rangle$ is the mean absolute magnitude of the star, defined as the apparent magnitude measured from a distance of $10 \mathrm{pc}$. The apparent and absolute magnitudes are related as $$ m-\mathfrak{M}=5 \log d-5 $$ where $d$ is the distance measured in pc. [figure1] Figure 1: Left: Colour composite view of the circumstellar nebula around RS Puppis assembled from HST images. Credit: NASA, ESA, and the Hubble Heritage Team (STScI/AURA)-Hubble/Europe Collaboration. Right: The light curve of RS Puppis. Phase 0 corresponds to the when the star is at its brightest, and the data has been folded over a known period of 41.5 days. Credit: AAVSO / P. Kervella et al. (2017). In the past, distances to stars could only be determined using something called the parallax method, where the angular shift of a star relative to the background stars was measured at two points in the Earth's orbit six months apart. The parallax angle is the same as the angle subtended by 1 au at the distance of the star. In Henrietta's time, parallax measurements only reached about $100 \mathrm{pc}$, however her method with Cepheid variables extended this range to more than $10^{6} \mathrm{pc}$, enabling much better determination of distances across the Milky Way and into nearby galaxies. The variable star RS Puppis (see Figure 1) is one of the biggest and brightest known Cepheids in the Milky Way galaxy and has one of the longest periods for this class of star at 41.5 days. In 2018 the Gaia satellite measured a parallax angle of 0.5844 milliarcseconds (where there are 3600 arcseconds in a degree). What does this suggest is the distance to the star?

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. problem: In the early 1900s, Henrietta Leavitt made several observations of variable stars from the Harvard College Observatory that led her to propose that the period of variation was related to the intrinsic brightness of the star. In particular, for a class of variable stars known as Cepheids, there was a strong power law relating luminosity to period, and hence a straight line on a log-log graph. Since magnitudes are $\propto \log L$, this can be described by the empirical relation $$ \langle\mathcal{M}\rangle=-2.43(\log P-1)-4.05 $$ where $P$ is the period measured in days and $\langle\mathcal{M}\rangle$ is the mean absolute magnitude of the star, defined as the apparent magnitude measured from a distance of $10 \mathrm{pc}$. The apparent and absolute magnitudes are related as $$ m-\mathfrak{M}=5 \log d-5 $$ where $d$ is the distance measured in pc. [figure1] Figure 1: Left: Colour composite view of the circumstellar nebula around RS Puppis assembled from HST images. Credit: NASA, ESA, and the Hubble Heritage Team (STScI/AURA)-Hubble/Europe Collaboration. Right: The light curve of RS Puppis. Phase 0 corresponds to the when the star is at its brightest, and the data has been folded over a known period of 41.5 days. Credit: AAVSO / P. Kervella et al. (2017). In the past, distances to stars could only be determined using something called the parallax method, where the angular shift of a star relative to the background stars was measured at two points in the Earth's orbit six months apart. The parallax angle is the same as the angle subtended by 1 au at the distance of the star. In Henrietta's time, parallax measurements only reached about $100 \mathrm{pc}$, however her method with Cepheid variables extended this range to more than $10^{6} \mathrm{pc}$, enabling much better determination of distances across the Milky Way and into nearby galaxies. The variable star RS Puppis (see Figure 1) is one of the biggest and brightest known Cepheids in the Milky Way galaxy and has one of the longest periods for this class of star at 41.5 days. In 2018 the Gaia satellite measured a parallax angle of 0.5844 milliarcseconds (where there are 3600 arcseconds in a degree). What does this suggest is the distance to the star? All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of pc, 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_161

由三颗星体构成的系统, 忽略其他星体对它们的作用, 存在着一种运动形式: 三颗星体在相互之间的万有引力作用下, 分别位于等边三角形的三个顶点上, 绕某一共同的圆心 $O$ 在三角形所在平面内以相同角速度做匀速圆周运动。如图所示, 三颗星体的质量均为 $m$, 三角形的边长为 $a$, 引力常量为 $G$, 下列说法正确的是（） [图1] A: 每个星体受到引力大小均为 $3 \frac{G m^{2}}{a^{2}}$ B: 每个星体的角速度均为 $\sqrt{\frac{3 G m}{a^{3}}}$ C: 若 $a$ 不变, $m$ 是原来的 2 倍, 则周期是原来的 $\frac{1}{2}$ D: 若 $m$ 不变, $a$ 是原来的 4 倍, 则线速度是原来的 $\frac{1}{2}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 由三颗星体构成的系统, 忽略其他星体对它们的作用, 存在着一种运动形式: 三颗星体在相互之间的万有引力作用下, 分别位于等边三角形的三个顶点上, 绕某一共同的圆心 $O$ 在三角形所在平面内以相同角速度做匀速圆周运动。如图所示, 三颗星体的质量均为 $m$, 三角形的边长为 $a$, 引力常量为 $G$, 下列说法正确的是（） [图1] A: 每个星体受到引力大小均为 $3 \frac{G m^{2}}{a^{2}}$ B: 每个星体的角速度均为 $\sqrt{\frac{3 G m}{a^{3}}}$ C: 若 $a$ 不变, $m$ 是原来的 2 倍, 则周期是原来的 $\frac{1}{2}$ D: 若 $m$ 不变, $a$ 是原来的 4 倍, 则线速度是原来的 $\frac{1}{2}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_339

1772 年, 法籍意大利数学家拉格朗日在论文《三体问题》中指出: 两个质量相差悬殊的天体 (如太阳和地球) 所在同一平面上有 5 个特殊点, 如图中 $L_{1} 、 L_{2} 、 L_{3} 、 L_{4}$ 、 $L_{5}$ 所示, 人们称之为拉格朗日点。若飞行器位于这些点上, 会在太阳与地球共同引力作用下, 可以不消耗燃料而保持与地球同步绕太阳做圆周运动。已知太阳质量为地球质量的 33 万倍, 日地距离为太阳半径的 215 倍, $L_{4}$ 和 $L_{5}$ 对称, 且与太阳连线的夹角为 $120^{\circ}$ ，则下列说法正确的是（） [图1] A: 人造卫星在 $L_{1}$ 和 $L_{2}$ 的向心加速度不同 B: $L_{3}$ 距太阳中心的距离比 $L_{2}$ 距太阳中心的距离大 C: $L_{1} 、 L_{2}$ 的公转线速度相同 D: $L_{4}$ 的旋转中心点在太阳内部

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: 1772 年, 法籍意大利数学家拉格朗日在论文《三体问题》中指出: 两个质量相差悬殊的天体 (如太阳和地球) 所在同一平面上有 5 个特殊点, 如图中 $L_{1} 、 L_{2} 、 L_{3} 、 L_{4}$ 、 $L_{5}$ 所示, 人们称之为拉格朗日点。若飞行器位于这些点上, 会在太阳与地球共同引力作用下, 可以不消耗燃料而保持与地球同步绕太阳做圆周运动。已知太阳质量为地球质量的 33 万倍, 日地距离为太阳半径的 215 倍, $L_{4}$ 和 $L_{5}$ 对称, 且与太阳连线的夹角为 $120^{\circ}$ ，则下列说法正确的是（） [图1] A: 人造卫星在 $L_{1}$ 和 $L_{2}$ 的向心加速度不同 B: $L_{3}$ 距太阳中心的距离比 $L_{2}$ 距太阳中心的距离大 C: $L_{1} 、 L_{2}$ 的公转线速度相同 D: $L_{4}$ 的旋转中心点在太阳内部 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_933

On $19^{\text {th }}$ July 2020 the first Arab interplanetary mission was launched on its way to Mars, consisting of the Hope probe designed by the United Arab Emirates Space Agency. When it reaches Mars in February 2021 it will do a short burn at a distance of $49400 \mathrm{~km}$ away from the surface to slow it down and put it into an elliptical capture orbit which will bring it as close as only $1000 \mathrm{~km}$ above the planet. Given the mass of Mars is $6.39 \times 10^{23} \mathrm{~kg}$ and its radius is $3390 \mathrm{~km}$, what will be the period of this orbit? A: 37.3 hours B: 40.9 hours C: 105 hours D: 116 hours

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 $19^{\text {th }}$ July 2020 the first Arab interplanetary mission was launched on its way to Mars, consisting of the Hope probe designed by the United Arab Emirates Space Agency. When it reaches Mars in February 2021 it will do a short burn at a distance of $49400 \mathrm{~km}$ away from the surface to slow it down and put it into an elliptical capture orbit which will bring it as close as only $1000 \mathrm{~km}$ above the planet. Given the mass of Mars is $6.39 \times 10^{23} \mathrm{~kg}$ and its radius is $3390 \mathrm{~km}$, what will be the period of this orbit? A: 37.3 hours B: 40.9 hours C: 105 hours D: 116 hours You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

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Astronomy

EN

text-only

Astronomy_440

我国的“嫦娥奔月”月球探测工程启动至今, 以“绕、落、回”为发展过程。中国国家航天局目前计划于 2020 年发射嫦娥工程第二阶段的月球车嫦娥四号。中国探月计划总工程师吴伟仁近期透露, 此台月球车很可能在离地球较远的月球背面着陆, 假设运载火箭先将 “嫦娥四号” 月球探测器成功送入太空, 由地月转移轨道进入半径为 $r_{l}=100$ 公里环月圆轨道I后成功变轨到近月点为 15 公里的椭圆轨道II, 在从 15 公里高度降至近月表面圆轨道III, 最后成功实现登月。若取两物体相距无穷远时的引力势能为零, 一个质量为 $m$ 的质点距质量为 $M$ 的引力中心为 $r$ 时, 其万有引力势能表达式为 $E_{\mathrm{P}}=-G \frac{M m}{r}$ （式中 $G$ 为引力常数）。已知月球质量 $M_{0}$, 月球半径为 $R$, 发射的“嫦娥四号”探测器质量 为 $m_{0}$, 引力常量 $G$ 。 则关于“嫦娥四号” 登月过程的说法正确的是 ( ) [图1] A: “嫦娥四号”探测器在轨道I上运行的动能大于在轨道III运行的动能 B: “嫦娥四号”探测器从轨道I上变轨到轨道III上时, 势能减小了 $G M_{0} m_{0}\left(\frac{1}{r_{1}}-\frac{1}{R}\right)$ C: “嫦娥四号”探测器在轨道III上运行时机械能等于在轨道I运行时的机械能 D: 落月的“嫦娥四号” 探测器从轨道III回到轨道I, 所要提供的最小能量是 $$ \frac{G M_{0} m}{2}\left(\frac{1}{R}-\frac{1}{r_{1}}\right) $$

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: 我国的“嫦娥奔月”月球探测工程启动至今, 以“绕、落、回”为发展过程。中国国家航天局目前计划于 2020 年发射嫦娥工程第二阶段的月球车嫦娥四号。中国探月计划总工程师吴伟仁近期透露, 此台月球车很可能在离地球较远的月球背面着陆, 假设运载火箭先将 “嫦娥四号” 月球探测器成功送入太空, 由地月转移轨道进入半径为 $r_{l}=100$ 公里环月圆轨道I后成功变轨到近月点为 15 公里的椭圆轨道II, 在从 15 公里高度降至近月表面圆轨道III, 最后成功实现登月。若取两物体相距无穷远时的引力势能为零, 一个质量为 $m$ 的质点距质量为 $M$ 的引力中心为 $r$ 时, 其万有引力势能表达式为 $E_{\mathrm{P}}=-G \frac{M m}{r}$ （式中 $G$ 为引力常数）。已知月球质量 $M_{0}$, 月球半径为 $R$, 发射的“嫦娥四号”探测器质量 为 $m_{0}$, 引力常量 $G$ 。 则关于“嫦娥四号” 登月过程的说法正确的是 ( ) [图1] A: “嫦娥四号”探测器在轨道I上运行的动能大于在轨道III运行的动能 B: “嫦娥四号”探测器从轨道I上变轨到轨道III上时, 势能减小了 $G M_{0} m_{0}\left(\frac{1}{r_{1}}-\frac{1}{R}\right)$ C: “嫦娥四号”探测器在轨道III上运行时机械能等于在轨道I运行时的机械能 D: 落月的“嫦娥四号” 探测器从轨道III回到轨道I, 所要提供的最小能量是 $$ \frac{G M_{0} m}{2}\left(\frac{1}{R}-\frac{1}{r_{1}}\right) $$ 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

ZH

multi-modal

Astronomy_923

Looking up into the UK sky at $10 \mathrm{pm}$ in late September, which of the following bright stars is NOT visible? A: Deneb (Right ascension $=20^{\mathrm{h}} 41^{\mathrm{m}}$, declination $\left.=+45.28^{\circ}\right)$ B: Vega (Right ascension $=18^{\mathrm{h}} 37^{\mathrm{m}}$, declination $\left.=+38.78^{\circ}\right)$ C: Capella (Right ascension $=05^{\mathrm{h}} 17^{\mathrm{m}}$, declination $=+46.00^{\circ}$ ) D: Sirius (Right ascension $=06^{\mathrm{h}} 45^{\mathrm{m}}$, declination $=-16.72^{\circ}$ )

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: Looking up into the UK sky at $10 \mathrm{pm}$ in late September, which of the following bright stars is NOT visible? A: Deneb (Right ascension $=20^{\mathrm{h}} 41^{\mathrm{m}}$, declination $\left.=+45.28^{\circ}\right)$ B: Vega (Right ascension $=18^{\mathrm{h}} 37^{\mathrm{m}}$, declination $\left.=+38.78^{\circ}\right)$ C: Capella (Right ascension $=05^{\mathrm{h}} 17^{\mathrm{m}}$, declination $=+46.00^{\circ}$ ) D: Sirius (Right ascension $=06^{\mathrm{h}} 45^{\mathrm{m}}$, declination $=-16.72^{\circ}$ ) You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D].

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Astronomy

EN

text-only

Astronomy_512

2019 年 4 月 10 日晚 9 时许, 全球多地天文学家同步公布的首张黑洞照片（如图）。假设银河系中两个黑洞 $\mathrm{A} 、 \mathrm{~B}$, 它们以二者连线上的 $O$ 点为圆心做匀速圆周运动, 测得 $\mathrm{A} 、 \mathrm{~B}$ 到 $O$ 点的距离分别为 $r$ 和 $2 r$ 。黑洞 $\mathrm{A} 、 \mathrm{~B}$ 均可看成质量分布均匀的球体，不考虑其他星球对黑洞的引力, 两黑洞的半径均远小于它们之间的距离。下列说法正确的是 A: 黑洞 $\mathrm{A} 、 \mathrm{~B}$ 的质量之比为 $1: 2$ B: 黑洞 A、B 所受引力之比为 $1: 2$ C: 黑洞 $\mathrm{A} 、 \mathrm{~B}$ 的角速度之比为 $1: 2$ D: 黑洞 $\mathrm{A} 、 \mathrm{~B}$ 的线速度之比为 $1: 2$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 2019 年 4 月 10 日晚 9 时许, 全球多地天文学家同步公布的首张黑洞照片（如图）。假设银河系中两个黑洞 $\mathrm{A} 、 \mathrm{~B}$, 它们以二者连线上的 $O$ 点为圆心做匀速圆周运动, 测得 $\mathrm{A} 、 \mathrm{~B}$ 到 $O$ 点的距离分别为 $r$ 和 $2 r$ 。黑洞 $\mathrm{A} 、 \mathrm{~B}$ 均可看成质量分布均匀的球体，不考虑其他星球对黑洞的引力, 两黑洞的半径均远小于它们之间的距离。下列说法正确的是 A: 黑洞 $\mathrm{A} 、 \mathrm{~B}$ 的质量之比为 $1: 2$ B: 黑洞 A、B 所受引力之比为 $1: 2$ C: 黑洞 $\mathrm{A} 、 \mathrm{~B}$ 的角速度之比为 $1: 2$ D: 黑洞 $\mathrm{A} 、 \mathrm{~B}$ 的线速度之比为 $1: 2$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

ZH

text-only

Astronomy_359

最近中国宇航局公布了天眼射电望远镜最新发现的一个行星系统, 该系统拥有一颗由岩石和气体构成的行星围绕一颗的类太阳恒星运行。经观测, 行星与恒星之间的距离是地、日间距离的 $\frac{1}{N}$, 恒星质量是太阳质量的 $k$ 倍，则下列叙述正确的是 ( ) A: 行星公转周期和地球公转周期的比值是 $N^{-\frac{3}{2}} k^{-\frac{1}{2}}$ B: 行星公转周期和地球公转周期的比值是 $N^{\frac{3}{2}} k^{\frac{1}{2}}$ C: 行星公转线速度和地球公转线速度的比值是 $N^{\frac{1}{2}} k^{\frac{1}{2}}$ D: 行星公转线速度和地球公转线速度的比值是 $N^{-\frac{1}{2}} k^{-\frac{1}{2}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 最近中国宇航局公布了天眼射电望远镜最新发现的一个行星系统, 该系统拥有一颗由岩石和气体构成的行星围绕一颗的类太阳恒星运行。经观测, 行星与恒星之间的距离是地、日间距离的 $\frac{1}{N}$, 恒星质量是太阳质量的 $k$ 倍，则下列叙述正确的是 ( ) A: 行星公转周期和地球公转周期的比值是 $N^{-\frac{3}{2}} k^{-\frac{1}{2}}$ B: 行星公转周期和地球公转周期的比值是 $N^{\frac{3}{2}} k^{\frac{1}{2}}$ C: 行星公转线速度和地球公转线速度的比值是 $N^{\frac{1}{2}} k^{\frac{1}{2}}$ D: 行星公转线速度和地球公转线速度的比值是 $N^{-\frac{1}{2}} k^{-\frac{1}{2}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

text-only

Astronomy_437

中国科幻大片《流浪地球 2 》中描述的“太空电梯”让人印象深刻。科学家们在地球同步轨道上建造了一个空间站, 再用超级缆绳连接地球赤道上的固定基地, 通过超级缆绳承载太空电梯, 使轿厢沿绳索从地球基地直入太空, 而向空间站运送货物。原理如图甲所示, 图中的太空电梯正停在离地面高 $R$ 处的站点修整, 并利用太阳能给蓄电池充电。图乙中 $r$ 为货物到地心的距离, $R$ 为地球半径, 曲线 $A$ 为地球引力对货物产生的加速度大小与 $r$ 的关系; 直线 $B$ 为货物由于地球自转而产生的向心加速度大小与 $r$ 的关系。关于相对地面静止在不同高度的电梯中货物, 下列说法正确的有（） [图1] [图2] 乙 A: 货物的线速度随着 $r$ 的增大而减小 B: 货物在 $r=R$ 处的线速度等于第一宇宙速度 C: 地球自转的周期 $T=2 \pi \sqrt{\frac{r_{0}}{a_{0}}}$ D: 图甲中质量为 $m$ 的货物对电梯底板的压力大小为 $\frac{m g_{0}}{4}-\frac{2 m r_{0} a_{0}}{R}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 中国科幻大片《流浪地球 2 》中描述的“太空电梯”让人印象深刻。科学家们在地球同步轨道上建造了一个空间站, 再用超级缆绳连接地球赤道上的固定基地, 通过超级缆绳承载太空电梯, 使轿厢沿绳索从地球基地直入太空, 而向空间站运送货物。原理如图甲所示, 图中的太空电梯正停在离地面高 $R$ 处的站点修整, 并利用太阳能给蓄电池充电。图乙中 $r$ 为货物到地心的距离, $R$ 为地球半径, 曲线 $A$ 为地球引力对货物产生的加速度大小与 $r$ 的关系; 直线 $B$ 为货物由于地球自转而产生的向心加速度大小与 $r$ 的关系。关于相对地面静止在不同高度的电梯中货物, 下列说法正确的有（） [图1] [图2] 乙 A: 货物的线速度随着 $r$ 的增大而减小 B: 货物在 $r=R$ 处的线速度等于第一宇宙速度 C: 地球自转的周期 $T=2 \pi \sqrt{\frac{r_{0}}{a_{0}}}$ D: 图甲中质量为 $m$ 的货物对电梯底板的压力大小为 $\frac{m g_{0}}{4}-\frac{2 m r_{0} a_{0}}{R}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

multi-modal

Astronomy_544

2019 年 4 月 10 日，人类首张黑洞“照片”问世.黑洞是爱因斯坦广义相对论预言存在的一种天体, 它具有的超强引力使光也无法逃离它的势力范围, 即黑洞的逃逸速度大于光速. 理论分析表明.星球的逃逸速度是其第一宇宙速度的 $\sqrt{2}$ 倍. 已知地球绕太阳公转的轨道半径约为 $1.5 \times 10^{11} \mathrm{~m}$, 公转周期约为 $3.15 \times 10^{7} \mathrm{~s}$, 假设太阳演变为黑洞, 它的半径最大为 $\left(\right.$ 太阳的质量不变, 光速 $\left.c=3.0 \times 10^{8} \mathrm{~m} / \mathrm{s}\right)$ A: $1 \mathrm{~km}$ B: $3 \mathrm{~km}$ C: $100 \mathrm{~km}$ D: $300 \mathrm{~km}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 2019 年 4 月 10 日，人类首张黑洞“照片”问世.黑洞是爱因斯坦广义相对论预言存在的一种天体, 它具有的超强引力使光也无法逃离它的势力范围, 即黑洞的逃逸速度大于光速. 理论分析表明.星球的逃逸速度是其第一宇宙速度的 $\sqrt{2}$ 倍. 已知地球绕太阳公转的轨道半径约为 $1.5 \times 10^{11} \mathrm{~m}$, 公转周期约为 $3.15 \times 10^{7} \mathrm{~s}$, 假设太阳演变为黑洞, 它的半径最大为 $\left(\right.$ 太阳的质量不变, 光速 $\left.c=3.0 \times 10^{8} \mathrm{~m} / \mathrm{s}\right)$ A: $1 \mathrm{~km}$ B: $3 \mathrm{~km}$ C: $100 \mathrm{~km}$ D: $300 \mathrm{~km}$ 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_115

2019 年诺贝尔物理学奖授予了三位天文学家, 以表彰他们对于人类对宇宙演化方面的了解所作的贡献。其中两位学者的贡献是首次发现地外行星, 其主要原理是恒星和其行星在引力作用下构成一个“双星系统”, 恒星在周期性运动时, 可通过观察其光谱的周期性变化知道其运动周期, 从而证实其附近存在行星。若观测到的某恒星运动周期为 $T$, 并测得该恒星与行星的距离为 $L$, 已知万有引力常量为 $G$, 则由这些物理量可以求得 ( ) A: 行星的质量 B: 恒星的质量 C: 恒星与行星的质量之和 D: 恒星与行星圆周运动的半径之比

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 2019 年诺贝尔物理学奖授予了三位天文学家, 以表彰他们对于人类对宇宙演化方面的了解所作的贡献。其中两位学者的贡献是首次发现地外行星, 其主要原理是恒星和其行星在引力作用下构成一个“双星系统”, 恒星在周期性运动时, 可通过观察其光谱的周期性变化知道其运动周期, 从而证实其附近存在行星。若观测到的某恒星运动周期为 $T$, 并测得该恒星与行星的距离为 $L$, 已知万有引力常量为 $G$, 则由这些物理量可以求得 ( ) A: 行星的质量 B: 恒星的质量 C: 恒星与行星的质量之和 D: 恒星与行星圆周运动的半径之比 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

text-only

Astronomy_290

2019 年 4 月 10 日, 天文学家召开全球新闻发布会, 宣布首次直接拍摄到黑洞的照片。黑洞是一种密度极大、引力极大的天体, 以至于光都无法逃逸（光速为 $c$ )。若黑洞的质量为 $M$, 半径为 $R$, 引力常量为 $G$, 其逃逸速度公式为 $v^{\prime}=\sqrt{\frac{2 G M}{R}}$ 。如果天文学家观测到一天体以速度 $v$ 绕某黑洞做半径为 $r$ 的匀速圆周运动, 则下列说法正确的有 A: $M=\frac{v^{2} r}{G}$ B: $M=G v^{2} r$ C: 该黑洞的最大半径为 $\frac{2 G M}{c^{2}}$ D: 该黑洞表面的重力加速度为 $\frac{G M}{R^{2}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 2019 年 4 月 10 日, 天文学家召开全球新闻发布会, 宣布首次直接拍摄到黑洞的照片。黑洞是一种密度极大、引力极大的天体, 以至于光都无法逃逸（光速为 $c$ )。若黑洞的质量为 $M$, 半径为 $R$, 引力常量为 $G$, 其逃逸速度公式为 $v^{\prime}=\sqrt{\frac{2 G M}{R}}$ 。如果天文学家观测到一天体以速度 $v$ 绕某黑洞做半径为 $r$ 的匀速圆周运动, 则下列说法正确的有 A: $M=\frac{v^{2} r}{G}$ B: $M=G v^{2} r$ C: 该黑洞的最大半径为 $\frac{2 G M}{c^{2}}$ D: 该黑洞表面的重力加速度为 $\frac{G M}{R^{2}}$ 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应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

text-only

Astronomy_367

利用金星凌日现象, 我们可以估算出地球与太阳之间的平均距离。日地平均距离也被定义为 1 个天文单位 (1A.U.), 是天文学中常用的距离单位。 金星轨道在地球轨道内侧, 某些特殊时刻, 地球、金星、太阳恰在一条直线上, 这时从地球上可以看到金星就像一个小黑点一样在太阳表面缓慢移动, 如图甲所示, 天文学称之为“金星凌日”。在地球上的不同地点, 比如图乙中的 $A 、 B$ 两点, 它们在同一时刻观察到的金星在日面上的位置是不同的，我们分别记为 $A^{\prime} 、 B^{\prime}$ 。 在 $A 、 B$ 两地分别同时拍摄金星凌日的照片, 然后将其重合起来观察, 如图丙所示。发现太阳的直径是两列轨迹之间距离 $n$ 倍。若已知太阳直径对地面观察者的张角 (亦称视直径) 为 $\alpha, \alpha$ 为很小的角, 可认为: $\sin \alpha=\tan \alpha=\alpha$, 请写出日地距离的表达式 (用 $l 、 k 、 n 、 \alpha$ 表示)。 [图1] 甲 [图2] 丙 [图3] 乙

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 利用金星凌日现象, 我们可以估算出地球与太阳之间的平均距离。日地平均距离也被定义为 1 个天文单位 (1A.U.), 是天文学中常用的距离单位。 金星轨道在地球轨道内侧, 某些特殊时刻, 地球、金星、太阳恰在一条直线上, 这时从地球上可以看到金星就像一个小黑点一样在太阳表面缓慢移动, 如图甲所示, 天文学称之为“金星凌日”。在地球上的不同地点, 比如图乙中的 $A 、 B$ 两点, 它们在同一时刻观察到的金星在日面上的位置是不同的，我们分别记为 $A^{\prime} 、 B^{\prime}$ 。 在 $A 、 B$ 两地分别同时拍摄金星凌日的照片, 然后将其重合起来观察, 如图丙所示。发现太阳的直径是两列轨迹之间距离 $n$ 倍。若已知太阳直径对地面观察者的张角 (亦称视直径) 为 $\alpha, \alpha$ 为很小的角, 可认为: $\sin \alpha=\tan \alpha=\alpha$, 请写出日地距离的表达式 (用 $l 、 k 、 n 、 \alpha$ 表示)。 [图1] 甲 [图2] 丙 [图3] 乙 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

EX

Astronomy

ZH

multi-modal

Astronomy_1093

It is often said that the Sun rises in the East and sets in the West, however this is only true twice a year at the equinoxes. In the Northern hemisphere, the Sun will rise northwards of East on the June solstice, and southwards of East on the December solstice; this is directly tied in with the varying length of day too, since the Sun either has a greater or shorter distance to travel across the sky (see Figure 1). [figure1] Figure 1: Left: The path of the Sun across the sky during the equinoxes and solstices, as viewed by an observer in the Northern hemisphere at a latitude of $\sim 40^{\circ}$. Credit: Daniel V. Schroeder / Weber State University. Right: The same idea but viewed from Iceland at a latitude of $65^{\circ}$, where by being so close to the Artic circle the day length can get close to 24 hours in June and almost no daylight in December. Credit: Kristn Bjarnadttir / University of Iceland. During the equinox, the Sun travels along the projection of the Earth's equator. In this question, we will assume a circular orbit for the Earth, and all angles will be calculated in degrees. A simple model for the vertical angle between the Sun and the horizon (known at the altitude), $h$, as a function of the bearing on the horizon, $A$ (measured clockwise from North, also called the azimuth), the latitude of the observer, $\phi$ (positive in Northern hemisphere, negative in Southern hemisphere), and the vertical angle of the Sun relative to the celestial equator (known as the solar declination), $\delta$, is given as: $$ h=-\left(90^{\circ}-\phi\right) \cos (A)+\delta $$ The solar declination can be considered to vary sinusoidally over the year, going from a maximum of $\delta=+23.44^{\circ}$ at the June solstice (roughly $21^{\text {st }}$ June) to a minimum of $\delta=-23.44^{\circ}$ on the December solstice (roughly $21^{\text {st }}$ December). It can be shown using spherical trigonometry that the precise model connecting $\delta, h, \phi$ and $A$ is: $$ \sin (\delta)=\sin (h) \sin (\phi)+\cos (h) \cos (\phi) \cos (A) . $$ Using the precise model, the path of the Sun across the sky forms a shape that is not quite the cosine shape of the simple model, and is shown in Figure 2. [figure2] Figure 2: The altitude of the Sun as a function of bearing during the equinoxes and solstices, as viewed by an observer at a latitude of $+56^{\circ}$. Whilst it resembles the cosine shape of the simple model well at this latitude, there are small deviations. Credit: Wikipedia. By using further spherical trigonometry, we can derive a second helpful equation in the precise model: $$ \sin (h)=\sin (\phi) \sin (\delta)+\cos (\phi) \cos (\delta) \cos (H) $$ Here, $H$ is the solar hour angle, which measures the angle between the Sun and solar noon as measured along the projection of the Earth's equator on the sky. Conventionally, $H=0^{\circ}$ at solar noon, is negative before solar noon, and is positive afterwards. Since the sun's hour angle increases at an approximately constant rate due to the rotation of the Earth, we can convert this angle into a time using the conversion $360^{\circ}=24^{\mathrm{h}}$.d. This exam is being taken on $24^{\text {th }}$ January and is 3 hours long. iii. What is the latitude with the longest sunrise today? Give its duration in minutes and seconds.

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: It is often said that the Sun rises in the East and sets in the West, however this is only true twice a year at the equinoxes. In the Northern hemisphere, the Sun will rise northwards of East on the June solstice, and southwards of East on the December solstice; this is directly tied in with the varying length of day too, since the Sun either has a greater or shorter distance to travel across the sky (see Figure 1). [figure1] Figure 1: Left: The path of the Sun across the sky during the equinoxes and solstices, as viewed by an observer in the Northern hemisphere at a latitude of $\sim 40^{\circ}$. Credit: Daniel V. Schroeder / Weber State University. Right: The same idea but viewed from Iceland at a latitude of $65^{\circ}$, where by being so close to the Artic circle the day length can get close to 24 hours in June and almost no daylight in December. Credit: Kristn Bjarnadttir / University of Iceland. During the equinox, the Sun travels along the projection of the Earth's equator. In this question, we will assume a circular orbit for the Earth, and all angles will be calculated in degrees. A simple model for the vertical angle between the Sun and the horizon (known at the altitude), $h$, as a function of the bearing on the horizon, $A$ (measured clockwise from North, also called the azimuth), the latitude of the observer, $\phi$ (positive in Northern hemisphere, negative in Southern hemisphere), and the vertical angle of the Sun relative to the celestial equator (known as the solar declination), $\delta$, is given as: $$ h=-\left(90^{\circ}-\phi\right) \cos (A)+\delta $$ The solar declination can be considered to vary sinusoidally over the year, going from a maximum of $\delta=+23.44^{\circ}$ at the June solstice (roughly $21^{\text {st }}$ June) to a minimum of $\delta=-23.44^{\circ}$ on the December solstice (roughly $21^{\text {st }}$ December). It can be shown using spherical trigonometry that the precise model connecting $\delta, h, \phi$ and $A$ is: $$ \sin (\delta)=\sin (h) \sin (\phi)+\cos (h) \cos (\phi) \cos (A) . $$ Using the precise model, the path of the Sun across the sky forms a shape that is not quite the cosine shape of the simple model, and is shown in Figure 2. [figure2] Figure 2: The altitude of the Sun as a function of bearing during the equinoxes and solstices, as viewed by an observer at a latitude of $+56^{\circ}$. Whilst it resembles the cosine shape of the simple model well at this latitude, there are small deviations. Credit: Wikipedia. By using further spherical trigonometry, we can derive a second helpful equation in the precise model: $$ \sin (h)=\sin (\phi) \sin (\delta)+\cos (\phi) \cos (\delta) \cos (H) $$ Here, $H$ is the solar hour angle, which measures the angle between the Sun and solar noon as measured along the projection of the Earth's equator on the sky. Conventionally, $H=0^{\circ}$ at solar noon, is negative before solar noon, and is positive afterwards. Since the sun's hour angle increases at an approximately constant rate due to the rotation of the Earth, we can convert this angle into a time using the conversion $360^{\circ}=24^{\mathrm{h}}$. problem: d. This exam is being taken on $24^{\text {th }}$ January and is 3 hours long. iii. What is the latitude with the longest sunrise today? Give its duration in minutes and seconds. 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{~min} 3 \mathrm{~s}, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

multi-modal

Astronomy_487

2018 年 12 月 8 日凌晨, 我国成功发射一枚火箭, 将“嫦娥四号”探测器送上了天空,历经 110 个小时的飞行后，在离月球仅 100 公里的距离完美“刹车”, 进入近月轨道运行; 12 月 30 日 8 时 55 分, “嫦娥四号”在环月轨道成功实施变轨控制, 顺利进入月球背面的预定着陆准备轨道; 2019 年 1 月 3 日 10 时 15 分北京航天飞行控制中心向“嫦娥四号”探测器发出着陆指令: 开启变推力发动机, 逐步将探测器的速度降到零, 并不断调整姿态,在距月面 100 米处悬停，选定相对平坦区域后缓慢垂直下降，实现了世界上首次在月球 背面软着陆。探测器在着陆过程中沿坚直方向运动, 设悬停前减速阶段变推力发动机的平均作用力为 $F$, 经过时间 $t$ 将探测器的速度由 $v$ 减小到 0 。已知探测器质量为 $m$, 在近月轨道做匀速圆周运动的周期为 $T$, 引力常量为 $G$, 月球可视为质量分布均匀的球体,着陆过程中“嫦娥四号”探测器质量不变。则通过以上数据可求得（） [图1] A: 月球表面的重力加速度为 $g_{\text {月 }}=\frac{F}{m}$ B: 月球的半径 $R=\left(\frac{(F t-m v) T^{2}}{4 m t \pi^{2}}\right)^{\frac{1}{3}}$ C: 月球的质量 $M=\frac{(F t-m v)^{3} T^{4}}{16 \pi^{4} G m^{3} t^{3}}$ D: 月球的平均密度 $\rho=\frac{G T^{2}}{3 \pi}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 2018 年 12 月 8 日凌晨, 我国成功发射一枚火箭, 将“嫦娥四号”探测器送上了天空,历经 110 个小时的飞行后，在离月球仅 100 公里的距离完美“刹车”, 进入近月轨道运行; 12 月 30 日 8 时 55 分, “嫦娥四号”在环月轨道成功实施变轨控制, 顺利进入月球背面的预定着陆准备轨道; 2019 年 1 月 3 日 10 时 15 分北京航天飞行控制中心向“嫦娥四号”探测器发出着陆指令: 开启变推力发动机, 逐步将探测器的速度降到零, 并不断调整姿态,在距月面 100 米处悬停，选定相对平坦区域后缓慢垂直下降，实现了世界上首次在月球 背面软着陆。探测器在着陆过程中沿坚直方向运动, 设悬停前减速阶段变推力发动机的平均作用力为 $F$, 经过时间 $t$ 将探测器的速度由 $v$ 减小到 0 。已知探测器质量为 $m$, 在近月轨道做匀速圆周运动的周期为 $T$, 引力常量为 $G$, 月球可视为质量分布均匀的球体,着陆过程中“嫦娥四号”探测器质量不变。则通过以上数据可求得（） [图1] A: 月球表面的重力加速度为 $g_{\text {月 }}=\frac{F}{m}$ B: 月球的半径 $R=\left(\frac{(F t-m v) T^{2}}{4 m t \pi^{2}}\right)^{\frac{1}{3}}$ C: 月球的质量 $M=\frac{(F t-m v)^{3} T^{4}}{16 \pi^{4} G m^{3} t^{3}}$ D: 月球的平均密度 $\rho=\frac{G T^{2}}{3 \pi}$ 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_207

继 2004 年开始在四川西昌发射“嫦娥系列”卫星开展探月工程后, 2020 年 7 月 23 日 12 时 41 分，我国首次火星探测任务“天问一号”在海南文昌航天发射场由长征 5 号运载火箭发射升空，直接送入地火转移轨道，开启了我国探测行星之旅。“天问一号”将飞行约 7 个月抵达火星, 并通过 2 至 3 个月的环绕飞行后着陆火星表面, 开展火星的表面形貌、土壤特性、物质成分、水冰、大气、电离层、磁场等科学探测。下列说法正确的是 ( ) A: “天问一号”选择在海南文昌发射，是为了在同等发射条件下，提高同型火箭的运载能力 B: 在地火转移轨道飞行时, “天问一号”是一颗人造行星, 与地球、火星共同绕太阳公转 C: 若“嫦娥一号”绕月与“天问一号”绕火星都可看做是圆周运动, 则它们的运行周期的平方和轨道半径的 3 次方的比值 $\frac{T^{2}}{R^{3}}$ 相等 D: “天问一号”绕火星运动, 必须在火星赤道平面, 并且周期与火星自转周期相同

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: 继 2004 年开始在四川西昌发射“嫦娥系列”卫星开展探月工程后, 2020 年 7 月 23 日 12 时 41 分，我国首次火星探测任务“天问一号”在海南文昌航天发射场由长征 5 号运载火箭发射升空，直接送入地火转移轨道，开启了我国探测行星之旅。“天问一号”将飞行约 7 个月抵达火星, 并通过 2 至 3 个月的环绕飞行后着陆火星表面, 开展火星的表面形貌、土壤特性、物质成分、水冰、大气、电离层、磁场等科学探测。下列说法正确的是 ( ) A: “天问一号”选择在海南文昌发射，是为了在同等发射条件下，提高同型火箭的运载能力 B: 在地火转移轨道飞行时, “天问一号”是一颗人造行星, 与地球、火星共同绕太阳公转 C: 若“嫦娥一号”绕月与“天问一号”绕火星都可看做是圆周运动, 则它们的运行周期的平方和轨道半径的 3 次方的比值 $\frac{T^{2}}{R^{3}}$ 相等 D: “天问一号”绕火星运动, 必须在火星赤道平面, 并且周期与火星自转周期相同 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

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Astronomy

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text-only

Astronomy_212

2021 年 4 月 29 日, 长征五号 $B$ 遥二运载火箭搭载中国载人航天工程中第一个空间站核心舱“天和核心舱”，在海南文昌航天发射场发射升空。最终“天和核心舱”顺利进入离地约 $400 \mathrm{~km}$ 高的预定圆轨道, 运行速率约为 $7.7 \mathrm{~km} / \mathrm{s}$, 宇航员 $24 \mathrm{~h}$ 内可以看到 16 次日出日落。已知万有引力常量 $G$, 根据以上信息能估算出 ( ) A: 地球的半径 B: 地球的质量 C: “天和核心舱”的质量 D: “天和核心舱”受到的地球引力

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 4 月 29 日, 长征五号 $B$ 遥二运载火箭搭载中国载人航天工程中第一个空间站核心舱“天和核心舱”，在海南文昌航天发射场发射升空。最终“天和核心舱”顺利进入离地约 $400 \mathrm{~km}$ 高的预定圆轨道, 运行速率约为 $7.7 \mathrm{~km} / \mathrm{s}$, 宇航员 $24 \mathrm{~h}$ 内可以看到 16 次日出日落。已知万有引力常量 $G$, 根据以上信息能估算出 ( ) A: 地球的半径 B: 地球的质量 C: “天和核心舱”的质量 D: “天和核心舱”受到的地球引力 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

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Astronomy

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text-only

Astronomy_1038

The NEOWISE telescope discovered a new comet on $27^{\text {th }}$ March 2020, later given the official designation C/2020 F3 NEOWISE. Although when first discovered it only had an apparent magnitude of 18.0, it would become sufficiently bright that it could be seen with the naked eye by observers throughout the northern hemisphere, and was one of the brightest comets since Hale-Bopp in 1997. [figure1] Figure 4: The comet C/2020 F3 NEOWISE as seen from the UK in late July. Credit: Alex Calverley. On its discovery date the comet was 1.702 au from the Earth and 2.089 au from the Sun, and at perihelion (when it was closest to the Sun) on $3^{\text {rd }}$ July 2020 it was only 0.294649 au from the Sun. Given the comet's orbit has an eccentricity of 0.999188 , estimate the year of its next perihelion. For a spherical object, the luminosity of the reflected light is a function of how much of the lit surface of the object we can see, known as the phase (for example crescent and gibbous phases of the moon). This correction is known as the phase factor, $p(\theta)$, such that the total power reflected is related to the total power incident as $P_{\text {ref }}=P_{\text {inc }} \times p(\theta)$. You are given that $$ p(\theta)=B\left[\left(1-\frac{\theta}{\pi}\right) \cos \theta+\frac{1}{\pi} \sin \theta\right] $$ where $B$ is a constant and $\theta$ is the phase angle, as defined in Fig 5, and is measured in radians (a measure of angle such that $2 \pi$ radians $=360^{\circ}$. [figure2] Figure 5: The phase angle is defined as the angle between the Earth and Sun as viewed from the object. Credit: Wikipedia.

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 NEOWISE telescope discovered a new comet on $27^{\text {th }}$ March 2020, later given the official designation C/2020 F3 NEOWISE. Although when first discovered it only had an apparent magnitude of 18.0, it would become sufficiently bright that it could be seen with the naked eye by observers throughout the northern hemisphere, and was one of the brightest comets since Hale-Bopp in 1997. [figure1] Figure 4: The comet C/2020 F3 NEOWISE as seen from the UK in late July. Credit: Alex Calverley. On its discovery date the comet was 1.702 au from the Earth and 2.089 au from the Sun, and at perihelion (when it was closest to the Sun) on $3^{\text {rd }}$ July 2020 it was only 0.294649 au from the Sun. Given the comet's orbit has an eccentricity of 0.999188 , estimate the year of its next perihelion. For a spherical object, the luminosity of the reflected light is a function of how much of the lit surface of the object we can see, known as the phase (for example crescent and gibbous phases of the moon). This correction is known as the phase factor, $p(\theta)$, such that the total power reflected is related to the total power incident as $P_{\text {ref }}=P_{\text {inc }} \times p(\theta)$. You are given that $$ p(\theta)=B\left[\left(1-\frac{\theta}{\pi}\right) \cos \theta+\frac{1}{\pi} \sin \theta\right] $$ where $B$ is a constant and $\theta$ is the phase angle, as defined in Fig 5, and is measured in radians (a measure of angle such that $2 \pi$ radians $=360^{\circ}$. [figure2] Figure 5: The phase angle is defined as the angle between the Earth and Sun as viewed from the object. Credit: Wikipedia. All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value.

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Astronomy

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multi-modal

Astronomy_199

为“照亮”“嫦娥四号”“驾临”月球背面之路, 一颗承载地月中转通信任务的中继卫星将在“嫦娥四号”发射前半年进入到地月拉格朗日点 $L_{2}$, 如图。在该点, 地球、月球和中继卫星始终位于同一直线上, 且中继卫星绕地球做圆周运动的周期与月球绕地球做圆周运动的周期相同，则（） [图1] A: 中继卫星绕地球做圆周运动的周期为一年 B: 中继卫星做圆周运动的向心力仅由地球提供 C: 中继卫星的线速度小于月球运动的线速度 D: 中继卫星的向心加速度大于月球运动的向心加速度

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 为“照亮”“嫦娥四号”“驾临”月球背面之路, 一颗承载地月中转通信任务的中继卫星将在“嫦娥四号”发射前半年进入到地月拉格朗日点 $L_{2}$, 如图。在该点, 地球、月球和中继卫星始终位于同一直线上, 且中继卫星绕地球做圆周运动的周期与月球绕地球做圆周运动的周期相同，则（） [图1] A: 中继卫星绕地球做圆周运动的周期为一年 B: 中继卫星做圆周运动的向心力仅由地球提供 C: 中继卫星的线速度小于月球运动的线速度 D: 中继卫星的向心加速度大于月球运动的向心加速度 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

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multi-modal

Astronomy_695

2019 年 3 月 10 日, 长征三号乙运载火箭将“中星 $6 \mathrm{C}$ ”通信卫星（记为卫星I）送入地 球同步轨道上, 主要为我国、东南亚、澳洲和南太平洋岛国等地区提供通信与广播业务。在同平面内的圆轨道上有一颗中轨道卫星II, 它运动的每个周期内都有一段时间 $t(t$ 未知）无法直接接收到卫星I发出的电磁波信号，因为其轨道上总有一段区域没有被卫星I 发出的电磁波信号覆盖到, 这段区域对应的圆心角为 $2 \alpha$ 。已知卫星I对地球的张角为 $2 \beta$,地球自转周期为 $T_{0}$, 万有引力常量为 $G$, 则根据题中条件, 可求出 ( ) [图1] A: 地球的平均密度为 $\frac{3 \pi}{G T_{0}{ }^{2}}$ B: 卫星I、II的角速度之比为 $\frac{\sin \beta}{\sin (\alpha-\beta)}$ C: 卫星II的周期为 $\sqrt{\frac{\sin ^{3} \beta}{\sin ^{3}(\alpha-\beta)}} \cdot T_{0}$ D: 题中时间 $t$ 为 $\sqrt{\frac{\sin ^{3} \beta}{\sin ^{3}(\alpha-\beta)}} \cdot \frac{\alpha}{\pi} T_{0}$

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: 2019 年 3 月 10 日, 长征三号乙运载火箭将“中星 $6 \mathrm{C}$ ”通信卫星（记为卫星I）送入地 球同步轨道上, 主要为我国、东南亚、澳洲和南太平洋岛国等地区提供通信与广播业务。在同平面内的圆轨道上有一颗中轨道卫星II, 它运动的每个周期内都有一段时间 $t(t$ 未知）无法直接接收到卫星I发出的电磁波信号，因为其轨道上总有一段区域没有被卫星I 发出的电磁波信号覆盖到, 这段区域对应的圆心角为 $2 \alpha$ 。已知卫星I对地球的张角为 $2 \beta$,地球自转周期为 $T_{0}$, 万有引力常量为 $G$, 则根据题中条件, 可求出 ( ) [图1] A: 地球的平均密度为 $\frac{3 \pi}{G T_{0}{ }^{2}}$ B: 卫星I、II的角速度之比为 $\frac{\sin \beta}{\sin (\alpha-\beta)}$ C: 卫星II的周期为 $\sqrt{\frac{\sin ^{3} \beta}{\sin ^{3}(\alpha-\beta)}} \cdot T_{0}$ D: 题中时间 $t$ 为 $\sqrt{\frac{\sin ^{3} \beta}{\sin ^{3}(\alpha-\beta)}} \cdot \frac{\alpha}{\pi} T_{0}$ 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]

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Astronomy

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multi-modal

Astronomy_383

2018 年 12 月 8 日, 嫦娥四号发射升空。将实现人类历史上首次月球背面登月。随着嫦娥奔月梦想的实现，我国不断刷新深空探测的中国高度。嫦娥卫星整个飞行过程可分为三个轨道段: 绕地飞行调相轨道段、地月转移轨道段、绕月飞行轨道段我们用如图所示的模型来简化描绘嫦娥卫星飞行过程, 假设调相轨道和绕月轨道的半长轴分别为 $a 、 b$, 公转周期分别为 $T_{1} 、 T_{2}$ 。关于嫦娥卫星的飞行过程, 下列说法正确的是（） [图1] A: $\frac{a^{3}}{T_{1}^{2}}=\frac{b^{3}}{T_{2}^{2}}$ B: 嫦娥卫星在地月转移轨道上运行的速度应大于 $11.2 \mathrm{~km} / \mathrm{s}$ C: 从调相轨道切入到地月转移轨道时, 卫星在 $P$ 点必须减速 D: 从地月转移轨道切入到绕月轨道时, 卫星在 $Q$ 点必须减速

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个单选题（只有一个正确答案）。 问题: 2018 年 12 月 8 日, 嫦娥四号发射升空。将实现人类历史上首次月球背面登月。随着嫦娥奔月梦想的实现，我国不断刷新深空探测的中国高度。嫦娥卫星整个飞行过程可分为三个轨道段: 绕地飞行调相轨道段、地月转移轨道段、绕月飞行轨道段我们用如图所示的模型来简化描绘嫦娥卫星飞行过程, 假设调相轨道和绕月轨道的半长轴分别为 $a 、 b$, 公转周期分别为 $T_{1} 、 T_{2}$ 。关于嫦娥卫星的飞行过程, 下列说法正确的是（） [图1] A: $\frac{a^{3}}{T_{1}^{2}}=\frac{b^{3}}{T_{2}^{2}}$ B: 嫦娥卫星在地月转移轨道上运行的速度应大于 $11.2 \mathrm{~km} / \mathrm{s}$ C: 从调相轨道切入到地月转移轨道时, 卫星在 $P$ 点必须减速 D: 从地月转移轨道切入到绕月轨道时, 卫星在 $Q$ 点必须减速 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

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Astronomy

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multi-modal

Astronomy_883

What is the perihelion of comet "SMukherjee2017"? A: 3.0 $\mathrm{AU}$ B: 3.5 AU C: 4.0 AU D: $4.5 \mathrm{AU}$ E: $5.0 \mathrm{AU}$

You are participating in an international Astronomy competition and need to solve the following question. This is a multiple choice question (only one correct answer). problem: What is the perihelion of comet "SMukherjee2017"? A: 3.0 $\mathrm{AU}$ B: 3.5 AU C: 4.0 AU D: $4.5 \mathrm{AU}$ E: $5.0 \mathrm{AU}$ You can solve it step by step. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER should be one of the options: [A, B, C, D, E].

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Astronomy

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text-only

Astronomy_531

宇宙中两颗相距较近的天体称为“双星”, 它们以两者连线上的某一点为圆心做匀速圆周运动, 而不至于因万有引力的作用吸引到一起. 设两者的质量分别为 $m_{1}$ 和 $m_{2}$ 且 $m_{1}>m_{2}$则下列说法正确的是( ) A: 两天体做圆周运动的周期相等 B: 两天体做圆周运动的向心加速度大小相等 C: $m_{1}$ 的轨道半径大于 $m_{2}$ 的轨道半径 D: $m_{2}$ 的轨道半径大于 $m_{l}$ 的轨道半径

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 宇宙中两颗相距较近的天体称为“双星”, 它们以两者连线上的某一点为圆心做匀速圆周运动, 而不至于因万有引力的作用吸引到一起. 设两者的质量分别为 $m_{1}$ 和 $m_{2}$ 且 $m_{1}>m_{2}$则下列说法正确的是( ) A: 两天体做圆周运动的周期相等 B: 两天体做圆周运动的向心加速度大小相等 C: $m_{1}$ 的轨道半径大于 $m_{2}$ 的轨道半径 D: $m_{2}$ 的轨道半径大于 $m_{l}$ 的轨道半径 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

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Astronomy

ZH

text-only

Astronomy_649

假设在半径为 $R$ 的某天体上发射一颗该天体的卫星, 已知引力常量为 $G$, 忽略该天体自转。 若卫星贴近该天体的表面做匀速圆周运动的周期为 $T_{2}$, 则该天体的密度是多少?

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

EX

Astronomy

ZH

text-only

Astronomy_223

地球赤道表面上的一物体质量为 $m_{1}$, 它相对地心的速度为 $v_{1}$, 地球同步卫星离地面的高度为 $h$, 它相对地心的速度为 $v_{2}$, 其质量为 $m_{2}$ 。已知地球的质量为 $M$, 半径为 $R$, 自转角速度为 $\omega$, 表面的重力加速度为 $g$, 地球的第一宇宙速度为 $v$, 万有引力常量为 $G$ 。下列各式成立的是 $(\quad)$ A: $v_{1}$ 小于 $v$ B: $\frac{v_{1}}{R}=\frac{v_{2}}{R+h}$ C: $m_{1} g=\frac{m_{1} v_{1}^{2}}{R}$ D: $\frac{v}{v_{2}}=\sqrt{\frac{R+h}{R}}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 地球赤道表面上的一物体质量为 $m_{1}$, 它相对地心的速度为 $v_{1}$, 地球同步卫星离地面的高度为 $h$, 它相对地心的速度为 $v_{2}$, 其质量为 $m_{2}$ 。已知地球的质量为 $M$, 半径为 $R$, 自转角速度为 $\omega$, 表面的重力加速度为 $g$, 地球的第一宇宙速度为 $v$, 万有引力常量为 $G$ 。下列各式成立的是 $(\quad)$ A: $v_{1}$ 小于 $v$ B: $\frac{v_{1}}{R}=\frac{v_{2}}{R+h}$ C: $m_{1} g=\frac{m_{1} v_{1}^{2}}{R}$ D: $\frac{v}{v_{2}}=\sqrt{\frac{R+h}{R}}$ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

text-only

Astronomy_581

在星球 $\mathrm{A}$ 上将一小物块 $P$ 坚直向上抛出, $P$ 的速度的二次方 $v^{2}$ 与位移 $x$ 间的关系如图中实线所示; 在另一星球 $B$ 上用另一小物块 $Q$ 完成同样的过程, $Q$ 的 $v^{2}-x$ 关系如图中虚线所示. 已知 $\mathrm{A}$ 的半径是 $B$ 的半径的 $\frac{1}{3}$, 若两星球均为质量均匀分布的球体 (球的体积公式为 $V=\frac{4}{3} \pi r^{3}, r$ 为球的半径), 两星球上均没有空气, 不考虑两星球的自转,则 ( ) [图1] A: A 表面的重力加速度是 $B$ 表面的重力加速度的 9 倍 B: $P$ 抛出后落回原处的时间是 $Q$ 抛出后落回原处的时间的 $\frac{1}{9}$ C: A 的密度是 $B$ 的密度的 9 倍 D: A 的第一宇宙速度是 $B$ 的第一宇宙速度的 $\sqrt{3}$ 倍

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 在星球 $\mathrm{A}$ 上将一小物块 $P$ 坚直向上抛出, $P$ 的速度的二次方 $v^{2}$ 与位移 $x$ 间的关系如图中实线所示; 在另一星球 $B$ 上用另一小物块 $Q$ 完成同样的过程, $Q$ 的 $v^{2}-x$ 关系如图中虚线所示. 已知 $\mathrm{A}$ 的半径是 $B$ 的半径的 $\frac{1}{3}$, 若两星球均为质量均匀分布的球体 (球的体积公式为 $V=\frac{4}{3} \pi r^{3}, r$ 为球的半径), 两星球上均没有空气, 不考虑两星球的自转,则 ( ) [图1] A: A 表面的重力加速度是 $B$ 表面的重力加速度的 9 倍 B: $P$ 抛出后落回原处的时间是 $Q$ 抛出后落回原处的时间的 $\frac{1}{9}$ C: A 的密度是 $B$ 的密度的 9 倍 D: A 的第一宇宙速度是 $B$ 的第一宇宙速度的 $\sqrt{3}$ 倍 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_638

人造地球卫星是发射数量最多, 用途最广, 发展最快的航天器。已知引力常量为 $G$, 地球半径为 $R$, 地球表面的重力加速度为 $\mathrm{g}$, 地球自转的周期为 $T$ 。 求地球同步卫星离地高度 $h_{l}$

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 人造地球卫星是发射数量最多, 用途最广, 发展最快的航天器。已知引力常量为 $G$, 地球半径为 $R$, 地球表面的重力加速度为 $\mathrm{g}$, 地球自转的周期为 $T$ 。 求地球同步卫星离地高度 $h_{l}$ 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

EX

Astronomy

ZH

text-only

Astronomy_1066

On Earth the Global Positioning System (GPS) requires a minimum of 24 satellites in orbit at any one time (there are typically more than that to allow for redundancies, with the current constellation having more than 30) so that at least 4 are visible above the horizon from anywhere on Earth (necessary for an $\mathrm{x}, \mathrm{y}, \mathrm{z}$ and time co-ordinate). This is achieved by having 6 different orbital planes, separated by $60^{\circ}$, and each orbital plane has 4 satellites. [figure1] Figure 1: The current set up of the GPS system used on Earth. Credits: Left: Peter H. Dana, University of Colorado; Right: GPS Standard Positioning Service Specification, $4^{\text {th }}$ edition The orbits are essentially circular with an eccentricity $<0.02$, an inclination of $55^{\circ}$, and an orbital period of exactly half a sidereal day (called a semi-synchronous orbit). The receiving angle of each satellite's antenna needs to be about $27.8^{\circ}$, and hence about $38 \%$ of the Earth's surface is within each satellite's footprint (see Figure 1), allowing the excellent coverage required. In the future we hope to colonise Mars, and so for navigation purposes it is likely that a type of GPS system will eventually be established on Mars too. Mars has a mass of $6.42 \times 10^{23} \mathrm{~kg}$, a mean radius of $3390 \mathrm{~km}$, a sidereal day of $24 \mathrm{~h} 37 \mathrm{mins}$, and two (low mass) moons with essentially circular orbits and semi-major axes of $9377 \mathrm{~km}$ (Phobos) and $23460 \mathrm{~km}$ (Deimos).c. Using suitable calculations, explore the viability of a 24-satellite GPS constellation similar to the one used on Earth, in a semi-synchronous Martian orbit, by considering: i. Would the moons prevent such an orbit?

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is a numerical value. Here is some context information for this question, which might assist you in solving it: On Earth the Global Positioning System (GPS) requires a minimum of 24 satellites in orbit at any one time (there are typically more than that to allow for redundancies, with the current constellation having more than 30) so that at least 4 are visible above the horizon from anywhere on Earth (necessary for an $\mathrm{x}, \mathrm{y}, \mathrm{z}$ and time co-ordinate). This is achieved by having 6 different orbital planes, separated by $60^{\circ}$, and each orbital plane has 4 satellites. [figure1] Figure 1: The current set up of the GPS system used on Earth. Credits: Left: Peter H. Dana, University of Colorado; Right: GPS Standard Positioning Service Specification, $4^{\text {th }}$ edition The orbits are essentially circular with an eccentricity $<0.02$, an inclination of $55^{\circ}$, and an orbital period of exactly half a sidereal day (called a semi-synchronous orbit). The receiving angle of each satellite's antenna needs to be about $27.8^{\circ}$, and hence about $38 \%$ of the Earth's surface is within each satellite's footprint (see Figure 1), allowing the excellent coverage required. In the future we hope to colonise Mars, and so for navigation purposes it is likely that a type of GPS system will eventually be established on Mars too. Mars has a mass of $6.42 \times 10^{23} \mathrm{~kg}$, a mean radius of $3390 \mathrm{~km}$, a sidereal day of $24 \mathrm{~h} 37 \mathrm{mins}$, and two (low mass) moons with essentially circular orbits and semi-major axes of $9377 \mathrm{~km}$ (Phobos) and $23460 \mathrm{~km}$ (Deimos). problem: c. Using suitable calculations, explore the viability of a 24-satellite GPS constellation similar to the one used on Earth, in a semi-synchronous Martian orbit, by considering: i. Would the moons prevent such an orbit? All mathematical formulas and symbols you output should be represented with LaTeX! You can solve it step by step. Remember, your answer should be calculated in the unit of \mathrm{~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_1194

In the heart of every star, nuclear fusion is taking place. For most stars that involves hydrogen being turned into helium, a process that starts by bringing two protons close enough that the strong nuclear force can act upon them. The smallest stars are the ones that have a core that is only just hot enough for fusion to occur, whilst in the biggest ones the radiation pressure of the photons given out by the fusion reaction pushing on the stellar material can overcome the gravitational forces holding it together. [figure1] Figure 5: Left: The lowest mass star we know of, EBLM J0555-57Ab, was found by von Boetticher et al. (2017) and is about the size of Saturn with a mass of $0.081 M_{\odot}$. Credit: Amanda Smith, University of Cambridge. Right: The highest mass star we know of, R136a1, is in the centre of the clump of stars on the right of this HST image of the Tarantula Nebula. Schneider et al. (2014) suggest it has a mass of $315 M_{\odot}$, which is above what stellar evolution models allow. Despite its large mass, other stars have far bigger radii. Credit: NASA \& ESA. For a spherical main sequence star made of a plasma (a fully ionized gas of electrons and nuclei) that is acting like an ideal gas, the temperature at the core can be approximately calculated as $$ T_{\mathrm{int}} \simeq \frac{G M \bar{\mu}}{k_{\mathrm{B}} R} \quad \text { where } \quad \bar{\mu}=\frac{m_{\mathrm{p}}}{2 X+3 Y / 4+Z / 2} . $$ In this equation, $M$ is the mass of the star, $R$ is its radius, $k_{\mathrm{B}}$ is the Boltzmann constant, and $\bar{\mu}$ is the mean mass of the plasma particles (i.e nuclei and electrons) with $m_{\mathrm{p}}$ the mass of a proton. Classically, the core of the Sun is not hot enough for fusion, and yet fusion is clearly happening. The key is that it is a fundamentally quantum process, and so protons are able to 'quantum tunnel' through the Coloumb barrier (see Figure 6), allowing fusion to occur at lower temperatures. In quantum mechanics, fusion will happen when $b=\lambda$ where $\lambda$ is the de Broglie wavelength of the proton, related to the momentum of the proton by $\lambda=h / p$. [figure2] Figure 6: A diagram showing the way a particle can pass through a classically impenetrable potential barrier due to its wave-like properties on the quantum scale. Credit: Brooks/Cole - Thomson Learning. In the smallest stars, electron degeneracy prevents them from compressing in radius and thus stops the core reaching $T_{\text {int }} \gtrsim T_{\text {quantum }}$. At the limit of electron degeneracy, the number density of electrons $n_{\mathrm{e}}=1 / \lambda_{\mathrm{e}}^{3}$ where $\lambda_{\mathrm{e}}$ is the de Broglie wavelength of the electrons. In the largest stars, radiation pressure pushes on the outer layers of the star stronger than gravity pulls them in. The brightest luminosity for a star is known as the Eddington luminosity, $L_{\text {Edd }}$. The acceleration due to radiation pressure can be calculated as $$ g_{\mathrm{rad}}=\frac{\kappa_{\mathrm{e}} I}{c} \quad \text { where } \quad \kappa_{\mathrm{e}}=\frac{\sigma_{\mathrm{T}}}{2 m_{\mathrm{p}}}(1+X) $$ and $\kappa_{\mathrm{e}}$ is the electron opacity of the stellar material, $\sigma_{\mathrm{T}}$ is the Thomson scattering cross-section for electrons $\left(=66.5 \mathrm{fm}^{2}\right.$ ), $X$ is the hydrogen fraction, and $I$ is the intensity of radiation (in $\mathrm{W} \mathrm{m}^{-2}$ ). Assuming main-sequence stars follow a mass-luminosity relation of $L \propto M^{3}$, the maximum mass of a star can be found by considering one that is radiating at $L_{\text {Edd }}$.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: In the heart of every star, nuclear fusion is taking place. For most stars that involves hydrogen being turned into helium, a process that starts by bringing two protons close enough that the strong nuclear force can act upon them. The smallest stars are the ones that have a core that is only just hot enough for fusion to occur, whilst in the biggest ones the radiation pressure of the photons given out by the fusion reaction pushing on the stellar material can overcome the gravitational forces holding it together. [figure1] Figure 5: Left: The lowest mass star we know of, EBLM J0555-57Ab, was found by von Boetticher et al. (2017) and is about the size of Saturn with a mass of $0.081 M_{\odot}$. Credit: Amanda Smith, University of Cambridge. Right: The highest mass star we know of, R136a1, is in the centre of the clump of stars on the right of this HST image of the Tarantula Nebula. Schneider et al. (2014) suggest it has a mass of $315 M_{\odot}$, which is above what stellar evolution models allow. Despite its large mass, other stars have far bigger radii. Credit: NASA \& ESA. For a spherical main sequence star made of a plasma (a fully ionized gas of electrons and nuclei) that is acting like an ideal gas, the temperature at the core can be approximately calculated as $$ T_{\mathrm{int}} \simeq \frac{G M \bar{\mu}}{k_{\mathrm{B}} R} \quad \text { where } \quad \bar{\mu}=\frac{m_{\mathrm{p}}}{2 X+3 Y / 4+Z / 2} . $$ In this equation, $M$ is the mass of the star, $R$ is its radius, $k_{\mathrm{B}}$ is the Boltzmann constant, and $\bar{\mu}$ is the mean mass of the plasma particles (i.e nuclei and electrons) with $m_{\mathrm{p}}$ the mass of a proton. Classically, the core of the Sun is not hot enough for fusion, and yet fusion is clearly happening. The key is that it is a fundamentally quantum process, and so protons are able to 'quantum tunnel' through the Coloumb barrier (see Figure 6), allowing fusion to occur at lower temperatures. In quantum mechanics, fusion will happen when $b=\lambda$ where $\lambda$ is the de Broglie wavelength of the proton, related to the momentum of the proton by $\lambda=h / p$. [figure2] Figure 6: A diagram showing the way a particle can pass through a classically impenetrable potential barrier due to its wave-like properties on the quantum scale. Credit: Brooks/Cole - Thomson Learning. In the smallest stars, electron degeneracy prevents them from compressing in radius and thus stops the core reaching $T_{\text {int }} \gtrsim T_{\text {quantum }}$. At the limit of electron degeneracy, the number density of electrons $n_{\mathrm{e}}=1 / \lambda_{\mathrm{e}}^{3}$ where $\lambda_{\mathrm{e}}$ is the de Broglie wavelength of the electrons. In the largest stars, radiation pressure pushes on the outer layers of the star stronger than gravity pulls them in. The brightest luminosity for a star is known as the Eddington luminosity, $L_{\text {Edd }}$. The acceleration due to radiation pressure can be calculated as $$ g_{\mathrm{rad}}=\frac{\kappa_{\mathrm{e}} I}{c} \quad \text { where } \quad \kappa_{\mathrm{e}}=\frac{\sigma_{\mathrm{T}}}{2 m_{\mathrm{p}}}(1+X) $$ and $\kappa_{\mathrm{e}}$ is the electron opacity of the stellar material, $\sigma_{\mathrm{T}}$ is the Thomson scattering cross-section for electrons $\left(=66.5 \mathrm{fm}^{2}\right.$ ), $X$ is the hydrogen fraction, and $I$ is the intensity of radiation (in $\mathrm{W} \mathrm{m}^{-2}$ ). Assuming main-sequence stars follow a mass-luminosity relation of $L \propto M^{3}$, the maximum mass of a star can be found by considering one that is radiating at $L_{\text {Edd }}$. 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 \mathrm{~K}, but when concluding your final answer, do not include the unit. Please end your response with: "The final answer is $\boxed{ANSWER}$", where ANSWER is the numerical value without any units.

NV

Astronomy

EN

multi-modal

Astronomy_307

如图, 赤道上空有 2 颗人造卫星 $\mathrm{A} 、 \mathrm{~B}$ 绕地球做同方向的匀速圆周运动, 地球半径为 $R$, 卫星 $\mathrm{A} 、 \mathrm{~B}$ 的轨道半径分别为 $\frac{5}{4} R 、 \frac{5}{3} R$, 卫星 $\mathrm{B}$ 的运动周期为 $T$, 某时刻 2 颗卫星与地心在同一直线上, 两颗卫星之间保持用光信号直接通信, 则（） [图1] A: 卫星 $\mathrm{A}$ 的加速度小于 $\mathrm{B}$ 的加速度 B: 卫星 $A 、 B$ 的周期之比为 $\frac{3 \sqrt{3}}{8}$ C: 再经时间 $t=\frac{3(8 \sqrt{3}+9) T}{148}$, 两颗卫星之间的通信将中断 D: 为了使赤道上任一点任一时刻均能接收到卫星 A 所在轨道的卫星的信号, 该轨道至少需要 3 颗卫星

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 如图, 赤道上空有 2 颗人造卫星 $\mathrm{A} 、 \mathrm{~B}$ 绕地球做同方向的匀速圆周运动, 地球半径为 $R$, 卫星 $\mathrm{A} 、 \mathrm{~B}$ 的轨道半径分别为 $\frac{5}{4} R 、 \frac{5}{3} R$, 卫星 $\mathrm{B}$ 的运动周期为 $T$, 某时刻 2 颗卫星与地心在同一直线上, 两颗卫星之间保持用光信号直接通信, 则（） [图1] A: 卫星 $\mathrm{A}$ 的加速度小于 $\mathrm{B}$ 的加速度 B: 卫星 $A 、 B$ 的周期之比为 $\frac{3 \sqrt{3}}{8}$ C: 再经时间 $t=\frac{3(8 \sqrt{3}+9) T}{148}$, 两颗卫星之间的通信将中断 D: 为了使赤道上任一点任一时刻均能接收到卫星 A 所在轨道的卫星的信号, 该轨道至少需要 3 颗卫星 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

MC

Astronomy

ZH

multi-modal

Astronomy_87

设想从地球赤道平面内架设一垂直于地面延伸到太空的电梯，电梯的箱体可以将人从地面运送到地球同步轨道的空间站。已知地球表面两极处的重力加速度为 $g$, 地球自转周期为 $T$, 地球半径为 $R$, 万有引力常量为 $G$ 。求同步轨道空间站距地面的高度 $h$;

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 设想从地球赤道平面内架设一垂直于地面延伸到太空的电梯，电梯的箱体可以将人从地面运送到地球同步轨道的空间站。已知地球表面两极处的重力加速度为 $g$, 地球自转周期为 $T$, 地球半径为 $R$, 万有引力常量为 $G$ 。求同步轨道空间站距地面的高度 $h$; 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

EX

Astronomy

ZH

text-only

Astronomy_412

“重力探矿” 是常用的探测黄金矿藏的方法之一, 是万有引力定律理论的实际应用, 其原理可简述如下: 如图, $P 、 Q$ 为某地区水平地面上的两点, 在 $P$ 点正下方一球形区域内充满了富含黄金的矿石，假定球形区域周围普通岩石均匀分布且密度为 $\rho$, 而球形区域内黄金矿石也均匀分布但其密度是普通岩石密度的 $(n+1)$ 倍, 如果没有这一球形区域黄金矿石的存在, 则该地区重力加速度（正常值）沿坚直方向，当该区域有黄金矿石时, 该地区重力加速度的大小和方向会与正常情况有微小偏离, 重力加速度在原坚直方向 (即 $P O$ 方向 $)$ 上的投影相对于正常值的偏离叫做“重力加速度反常”, 为了探寻黄金 矿石区域的位置和储量, 常利用 $P$ 点附近重力加速度反常现象, 已知引力常数为 $G$ 。若在水平地面上以 $P$ 点为圆心、半径为 $L$ 的范围内发现: 重力加速度反常值在 $\delta$与 $k \delta(k>1)$ 之间变化, 且重力加速度反常的最大值出现在 $P$ 点, 如果这种反常是由于地下存在某一球形区域黄金矿石造成的, 试求此球形区域球心的深度和球形区域的体积。 [图1]

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题包含多个待求解的量。 问题: “重力探矿” 是常用的探测黄金矿藏的方法之一, 是万有引力定律理论的实际应用, 其原理可简述如下: 如图, $P 、 Q$ 为某地区水平地面上的两点, 在 $P$ 点正下方一球形区域内充满了富含黄金的矿石，假定球形区域周围普通岩石均匀分布且密度为 $\rho$, 而球形区域内黄金矿石也均匀分布但其密度是普通岩石密度的 $(n+1)$ 倍, 如果没有这一球形区域黄金矿石的存在, 则该地区重力加速度（正常值）沿坚直方向，当该区域有黄金矿石时, 该地区重力加速度的大小和方向会与正常情况有微小偏离, 重力加速度在原坚直方向 (即 $P O$ 方向 $)$ 上的投影相对于正常值的偏离叫做“重力加速度反常”, 为了探寻黄金 矿石区域的位置和储量, 常利用 $P$ 点附近重力加速度反常现象, 已知引力常数为 $G$ 。若在水平地面上以 $P$ 点为圆心、半径为 $L$ 的范围内发现: 重力加速度反常值在 $\delta$与 $k \delta(k>1)$ 之间变化, 且重力加速度反常的最大值出现在 $P$ 点, 如果这种反常是由于地下存在某一球形区域黄金矿石造成的, 试求此球形区域球心的深度和球形区域的体积。 [图1] 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你的最终解答的量应该按以下顺序输出：[此球形区域球心的深度, 此球形区域球心的体积] 它们的答案类型依次是[表达式, 表达式] 你需要在输出的最后用以下格式总结答案：“最终答案是\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5

MPV

Astronomy

ZH

multi-modal

Astronomy_725

2023 年春节期间, 中国科幻电影《流浪地球 2 》热播。假设地球逃离太阳系过程如图所示, 地球现在绕太阳在圆轨道I上运行, 运动到 $A$ 点加速变轨进入椭圆轨道II, 在粗圆轨道II上运动到远日点 $B$ 时再次加速变轨, 从而摆脱太阳的束缚, 则地球 $(\quad)$ [图1] A: 在轨道I通过 $A$ 点的速度小于在轨道II通过 $A$ 点的速度 B: 沿轨道II运行时, 在 $A$ 点的加速度大于在 $B$ 点的加速度 C: 沿轨道II运行时, 由 $A$ 点运动到 $B$ 点的过程中, 动能逐渐增大 D: 沿轨道I和轨道II运行时, 相同时间内地球与太阳连线扫过的面积相等

你正在参加一个国际天文竞赛，并需要解决以下问题。 这是一个多选题（有多个正确答案）。 问题: 2023 年春节期间, 中国科幻电影《流浪地球 2 》热播。假设地球逃离太阳系过程如图所示, 地球现在绕太阳在圆轨道I上运行, 运动到 $A$ 点加速变轨进入椭圆轨道II, 在粗圆轨道II上运动到远日点 $B$ 时再次加速变轨, 从而摆脱太阳的束缚, 则地球 $(\quad)$ [图1] A: 在轨道I通过 $A$ 点的速度小于在轨道II通过 $A$ 点的速度 B: 沿轨道II运行时, 在 $A$ 点的加速度大于在 $B$ 点的加速度 C: 沿轨道II运行时, 由 $A$ 点运动到 $B$ 点的过程中, 动能逐渐增大 D: 沿轨道I和轨道II运行时, 相同时间内地球与太阳连线扫过的面积相等 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]

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Astronomy

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multi-modal

Astronomy_628

质量为 $m$ 的宇宙飞船, 在离月球地面高度 $h$ 处沿圆形轨道绕月球运行。为使飞船到达月球表面 $B$ 点, 喷气发动机在 $A$ 点做一次极短时间的喷气。从喷口射出的气流方向与圆周轨道相切且相对飞船的速度为 $u$, 月球半径为 $R, h=\frac{R}{16}, A 、 B$ 两点与球心在一直线上, 其速度与飞船到球心的距离成反比。月球表面重力加速度为 $g$ 。若以无穷远处为飞船引力势能的零势能点, 飞船和球心距离为 $r$ 时, 引力势能的表达式为 $E_{\mathrm{P}}=-\frac{G M m}{r}$ 。 ( $M$ 是月球质量, 本题中未知)。求: 飞船在圆形轨道上运行的速度大小 $v_{A}$; [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: 质量为 $m$ 的宇宙飞船, 在离月球地面高度 $h$ 处沿圆形轨道绕月球运行。为使飞船到达月球表面 $B$ 点, 喷气发动机在 $A$ 点做一次极短时间的喷气。从喷口射出的气流方向与圆周轨道相切且相对飞船的速度为 $u$, 月球半径为 $R, h=\frac{R}{16}, A 、 B$ 两点与球心在一直线上, 其速度与飞船到球心的距离成反比。月球表面重力加速度为 $g$ 。若以无穷远处为飞船引力势能的零势能点, 飞船和球心距离为 $r$ 时, 引力势能的表达式为 $E_{\mathrm{P}}=-\frac{G M m}{r}$ 。 ( $M$ 是月球质量, 本题中未知)。求: 飞船在圆形轨道上运行的速度大小 $v_{A}$; [图1] 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

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Astronomy

ZH

multi-modal

Astronomy_203

预计于 2022 年建成的中国空间站将成为中国空间和新技术研究的重要基地。假设空间站中的宇航员利用电热器对食品加热, 电热器的加热线圈可以简化成如图甲所示的圆形闭合线圈, 其匝数为 $n$, 半径为 $r_{1}$, 总电阻为 $R_{0}$ 。将此线圈垂直放在圆形磁场中,且保证两圆心重合, 圆形磁场的半径为 $r_{2}\left(r_{2}>r_{1}\right)$, 磁感应强度大小 $B$ 随时间 $t$ 的变化关系如图乙所示。求: 若已知空间站绕地球做匀速圆周运动的周期为 $T$, 地球的半径为 $R$ ，地球表面的重力加速度为 $g$, 则该空间站距离地球表面的高度 $h$ 为多少? (用所给物理量符号表示) [图1] 图甲 [图2] 图乙

你正在参加一个国际天文竞赛，并需要解决以下问题。 这个问题的答案是一个表达式。 问题: 预计于 2022 年建成的中国空间站将成为中国空间和新技术研究的重要基地。假设空间站中的宇航员利用电热器对食品加热, 电热器的加热线圈可以简化成如图甲所示的圆形闭合线圈, 其匝数为 $n$, 半径为 $r_{1}$, 总电阻为 $R_{0}$ 。将此线圈垂直放在圆形磁场中,且保证两圆心重合, 圆形磁场的半径为 $r_{2}\left(r_{2}>r_{1}\right)$, 磁感应强度大小 $B$ 随时间 $t$ 的变化关系如图乙所示。求: 若已知空间站绕地球做匀速圆周运动的周期为 $T$, 地球的半径为 $R$ ，地球表面的重力加速度为 $g$, 则该空间站距离地球表面的高度 $h$ 为多少? (用所给物理量符号表示) [图1] 图甲 [图2] 图乙 你输出的所有数学公式和符号应该使用LaTeX表示！ 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\frac{1}{2} g t^2

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Astronomy

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multi-modal

Astronomy_639

如图, 三个质点 $a 、 b 、 c$ 质量分别为 $m_{1} 、 m_{2} 、 M\left(M \gg m_{1}, M \gg m_{2}\right)$ 。在 $c$ 的万有引力作用下, $a 、 b$ 在同一平面内绕 $c$ 沿逆时针方向做匀速圆周运动, 它们的周期之比 $T_{a}$ : $T_{b}=1: 8, a 、 b$ 的轨道半径分别为 $r_{a}$ 和 $r_{b}$ 。从图示位置开始, 在 $b$ 运动一周的过程中,则 ( ) [图1] A: $r_{a}: r_{b}=1: 4$ B: B. $a 、 b$ 距离最近的次数为 8 次 C: $a 、 b$ 距离最远的次数为 9 次 D: $a 、 b 、 c$ 共线的次数为 16 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: 如图, 三个质点 $a 、 b 、 c$ 质量分别为 $m_{1} 、 m_{2} 、 M\left(M \gg m_{1}, M \gg m_{2}\right)$ 。在 $c$ 的万有引力作用下, $a 、 b$ 在同一平面内绕 $c$ 沿逆时针方向做匀速圆周运动, 它们的周期之比 $T_{a}$ : $T_{b}=1: 8, a 、 b$ 的轨道半径分别为 $r_{a}$ 和 $r_{b}$ 。从图示位置开始, 在 $b$ 运动一周的过程中,则 ( ) [图1] A: $r_{a}: r_{b}=1: 4$ B: B. $a 、 b$ 距离最近的次数为 8 次 C: $a 、 b$ 距离最远的次数为 9 次 D: $a 、 b 、 c$ 共线的次数为 16 次 你可以一步一步来解决这个问题，并输出详细的解答过程。 你需要在输出的最后用以下格式总结答案：“最终答案是$\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]

SC

Astronomy

ZH

multi-modal

Astronomy_910

Given the mass and radius of an exoplanet we can determine its likely composition, since the denser the material the smaller the planet will be for a given mass (see Figure 3). At a minimum orbital distance, called the Roche limit, tidal forces will cause a planet to break up, so very short period exoplanets place meaningful constraints on their composition as they must be very dense to survive in that orbit. The most extreme variant is called an 'iron planet', as it is made of iron with little or no rocky mantle ( $\gtrsim 80 \%$ iron by mass). The exoplanet KOI 1843.03 has the shortest known orbital period of a little over 4 hours, so is a potential candidate to be an iron planet. [figure1] Figure 3: Left: Comparisons of sizes of planets with different compositions [Marc Kuchner / NASA GSFC]. Right: An artist's impression of what an extreme iron planet (almost $\sim 100 \%$ iron) might look like. The Roche limiting distance, $a_{\min }$, for a body comprised of an incompressible fluid with negligible bulk tensile strength in a circular orbit about its parent star is $$ a_{\min }=2.44 R_{\star}\left(\frac{\rho_{\star}}{\rho_{p}}\right)^{1 / 3} $$ where $R_{\star}$ is the radius of the star, $\rho_{\star}$ is the density of the star and $\rho_{p}$ is the density of the planet. Over the mass range $0.1 M_{E}-1.0 M_{E}$ the mass-radius relation for pure silicate and pure iron planets is approximately a power law and can be described as $$ \log _{10}\left(\frac{R}{R_{E}}\right)=0.295 \log _{10}\left(\frac{M}{M_{E}}\right)+\alpha $$ where $\alpha=0.0286$ in the pure silicate case and $\alpha=-0.1090$ in the pure iron case. Derive a formula for the period of an exoplanet orbiting at the Roche limit, and hence show it is dependent only on the variable $\rho_{p}$ (i.e. independent of the properties of the parent star).

You are participating in an international Astronomy competition and need to solve the following question. The answer to this question is an expression. problem: Given the mass and radius of an exoplanet we can determine its likely composition, since the denser the material the smaller the planet will be for a given mass (see Figure 3). At a minimum orbital distance, called the Roche limit, tidal forces will cause a planet to break up, so very short period exoplanets place meaningful constraints on their composition as they must be very dense to survive in that orbit. The most extreme variant is called an 'iron planet', as it is made of iron with little or no rocky mantle ( $\gtrsim 80 \%$ iron by mass). The exoplanet KOI 1843.03 has the shortest known orbital period of a little over 4 hours, so is a potential candidate to be an iron planet. [figure1] Figure 3: Left: Comparisons of sizes of planets with different compositions [Marc Kuchner / NASA GSFC]. Right: An artist's impression of what an extreme iron planet (almost $\sim 100 \%$ iron) might look like. The Roche limiting distance, $a_{\min }$, for a body comprised of an incompressible fluid with negligible bulk tensile strength in a circular orbit about its parent star is $$ a_{\min }=2.44 R_{\star}\left(\frac{\rho_{\star}}{\rho_{p}}\right)^{1 / 3} $$ where $R_{\star}$ is the radius of the star, $\rho_{\star}$ is the density of the star and $\rho_{p}$ is the density of the planet. Over the mass range $0.1 M_{E}-1.0 M_{E}$ the mass-radius relation for pure silicate and pure iron planets is approximately a power law and can be described as $$ \log _{10}\left(\frac{R}{R_{E}}\right)=0.295 \log _{10}\left(\frac{M}{M_{E}}\right)+\alpha $$ where $\alpha=0.0286$ in the pure silicate case and $\alpha=-0.1090$ in the pure iron case. Derive a formula for the period of an exoplanet orbiting at the Roche limit, and hence show it is dependent only on the variable $\rho_{p}$ (i.e. independent of the properties of the parent star). 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