2024年IOAA理论第10题-最大日食 - 版本历史

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Quan787：创建页面，内容为“{{由ai生成}} ==英文题目== '''T10. Greatest Eclipse (75 points)''' The greatest eclipse is defined as the instant when the axis of the Moon's shadow cone get…”

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创建页面，内容为“{{由ai生成}} ==英文题目== '''T10. Greatest Eclipse (75 points)''' The greatest eclipse is defined as the instant when the axis of the Moon's shadow cone get…”

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{{由ai生成}}
==英文题目==

'''T10. Greatest Eclipse (75 points)'''

The greatest eclipse is defined as the instant when the axis of the Moon's shadow cone gets closest to the centre of the Earth in a solar eclipse. This problem explores the geometry of this phenomenon, using the solar eclipse of 29$^\text{th}$ May 1919 as an example, as it has great historical significance for being the first time astronomers were able to observationally verify general relativity. One of the scientific expeditions to observe this eclipse took place in the Brazilian city of Sobral.

The two following tables show the Cartesian and spherical coordinates of the Sun and the Moon at the time of the greatest eclipse. The system used for these coordinates is right-handed and has the origin at the centre of the Earth, the positive $x$-axis pointing towards the Greenwich meridian, and the positive $z$-axis pointing towards the North Pole. For the rest of this problem, this will be referred to as **System I**.

'''Spherical Coordinates:'''
{| class="wikitable"
|-
! !! Centre of the Sun !! Centre of the Moon
|-
! Radial Distance ($r$)
| $1.516 \times 10^{11}$ m || $3.589 \times 10^{8}$ m
|-
! Polar Angle ($\theta$)
| $68^\circ29'44.1''$ || $68^\circ47'41.6''$
|-
! Azimuthal Angle ($\varphi$)
| $-1^h11^m28.2^s$ || $-1^h11^m22.9^s$
|}

'''Cartesian Coordinates:'''
{| class="wikitable"
|-
! !! Centre of the Sun !! Centre of the Moon
|-
! $x$
| $1.342 \times 10^{11}$ m || $3.185 \times 10^{8}$ m
|-
! $y$
| $-4.327 \times 10^{10}$ m || $-1.025 \times 10^{8}$ m
|-
! $z$
| $5.557 \times 10^{10}$ m || $1.298 \times 10^{8}$ m
|}

==中文翻译==

'''T10. 最大食分（75分）'''

最大食分定义为月球本影锥轴最接近地球中心的时刻。本问题以1919年5月29日日食为例探讨该现象的几何学，此次日食因首次通过观测验证广义相对论而具有重大历史意义。科学考察队之一在巴西索布拉尔市观测了此次日食。

下表展示了最大食分时刻太阳和月球的球面坐标与笛卡尔坐标。坐标系为右手系，原点在地球中心，$x$轴正向指向格林尼治子午线，$z$轴正向指向北极点。下文中称此坐标系为**系统I**。

'''球面坐标系：'''
{| class="wikitable"
|-
! !! 太阳中心 !! 月球中心
|-
! 径向距离（$r$）
| $1.516 \times 10^{11}$ 米 || $3.589 \times 10^{8}$ 米
|-
! 极角（$\theta$）
| $68^\circ29'44.1''$ || $68^\circ47'41.6''$
|-
! 方位角（$\varphi$）
| $-1^h11^m28.2^s$ || $-1^h11^m22.9^s$
|}

'''笛卡尔坐标系：'''
{| class="wikitable"
|-
! !! 太阳中心 !! 月球中心
|-
! $x$
| $1.342 \times 10^{11}$ 米 || $3.185 \times 10^{8}$ 米
|-
! $y$
| $-4.327 \times 10^{10}$ 米 || $-1.025 \times 10^{8}$ 米
|-
! $z$
| $5.557 \times 10^{10}$ 米 || $1.298 \times 10^{8}$ 米
|}

==官方解答==

=== Part I: 地理坐标 ===
(a) 太阳和月球的赤纬：
$$$\delta = 90^\circ - \theta$$$
$$$\delta_\odot = 21^\circ30'15.9''$$$
$$$\delta_{Moon} = 21^\circ12'18.4''$$$

(b) 太阳和月球的赤经（格林尼治恒星时为$5^h32^m35.5^s$）：
$$$\alpha = \varphi + LST_{Greenwich}$$$
$$$\alpha_\odot = 4^h21^m7.3^s$$$
$$$\alpha_{Moon} = 4^h21^m12.6^s$$$

(c) 月球本影轴方向单位向量：
$$$\vec{u} = \left\langle -0.8855, 0.2855, -0.3666 \right\rangle$$$

(d) 本影轴与地球表面交点坐标：
通过方程$$|\vec{M} + k\vec{u}| = R_\oplus$$解得：
$$$\phi = 4^\circ22'N$$$
$$$\lambda = 16^\circ42'W$$$

=== Part II: 全食持续时间 ===
(e) 本影半径估算：
$$$r_{umbra} = 1.196 \times 10^5 \, \text{m}$$$

(f) 地球自转线速度：
$$$v_{rot} = 463.7 \, \text{m/s}$$$

(g) 月球轨道速度（忽略半长轴变化）：
$$$v_{Moon} = 1088 \, \text{m/s}$$$

(h) 系统II中月球速度向量：
$$$\mathbf{v_{II}} = \begin{bmatrix} 1085 \, \text{m/s} \\ 81.25 \, \text{m/s} \\ 0 \end{bmatrix}$$$

(i) 系统III中月球速度向量：
$$$\mathbf{v_{III}} = \begin{bmatrix} 1085 \, \text{m/s} \\ 77.76 \, \text{m/s} \\ -23.54 \, \text{m/s} \end{bmatrix}$$$

(j) 本影中心地表移动速度：
$$$v_{umbra} = \sqrt{621.4^2 + 77.76^2} = 626.3 \, \text{m/s}$$$

(k) 全食持续时间：
$$$\Delta t = \frac{2r_{umbra}}{v_{umbra}} = 6\,\text{分钟}\,21.4\,\text{秒}$$$

[[分类:日月食]]
[[分类:球面三角]]
[[分类:由ai生成]]

@@ 第1行： / 第1行： @@
 {{由ai生成}}
 ==英文题目==
@@ 第8行： / 第9行： @@
 The two following tables show the Cartesian and spherical coordinates of the Sun and the Moon at the time of the greatest eclipse. The system used for these coordinates is right-handed and has the origin at the centre of the Earth, the positive \(x\)-axis pointing towards the Greenwich meridian, and the positive \(z\)-axis pointing towards the North Pole. For the rest of this problem, this will be referred to as System I.
-Spherical Coordinates:
+'''Spherical Coordinates:'''
-|    | Centre of the Sun    | Centre of the Moon    |
+{| class="wikitable"
-|---|---|---|
+|-
-| Radial Distance (\(r\))   | \(1.516 \times 10^{11}\) m    | \(3.589 \times 10^8\) m    |
+! !! Centre of the Sun !! Centre of the Moon
-| Polar Angle (\(\theta\))   | \(68^{\circ}29'44.1''\)    | \(68^{\circ}47'41.6''\)    |
+|-
-| Azimuthal Angle (\(\varphi\)) | \(-1^h11^{m}28.2^s\)    | \(-1^h11^{m}22.9^s\)    |
+| Radial Distance (\(r\)) || \(1.516 \times 10^{11}\) m || \(3.589 \times 10^8\) m
-Cartesian Coordinates:
+'''Cartesian Coordinates:'''
-|    | Centre of the Sun    | Centre of the Moon    |
+{| class="wikitable"
-|---|---|---|
+|-
-| \(x\)    | \(1.342 \times 10^{11}\) m    | \(3.185 \times 10^8\) m    |
+! !! Centre of the Sun !! Centre of the Moon
-| \(y\)    | \(-4.327 \times 10^{10}\) m    | \(-1.025 \times 10^8\) m    |
+|-
-| \(z\)    | \(5.557 \times 10^{10}\) m    | \(1.298 \times 10^8\) m    |
+| \(x\) || \(1.342 \times 10^{11}\) m || \(3.185 \times 10^8\) m
 For this problem, assume that the Earth is a perfect sphere.
-Note: The spherical coordinates of a point \(P\) are defined as follows:
+'''Note:''' The spherical coordinates of a point \(P\) are defined as follows:
 - Radial distance (\(r\)): distance between the origin (\(O\)) and \(P\) (range: \(r \geq 0\)).
 - Polar angle (\(\theta\)): angle between the positive \(z\)-axis and the line segment \(OP\) (range: \(0^{\circ} \leq \theta \leq 180^{\circ}\)).
 - Azimuthal angle (\(\varphi\)): angle between the positive \(x\)-axis and the projection of the line segment \(OP\) onto the \(xy\)-plane (range: \(-12^h \leq \varphi < 12^h\)).
-===== Part I: Geographic Coordinates (25 points) =====
+=== Part I: Geographic Coordinates (25 points) ===
 (a) (3 points) Determine the declination of the Sun and the Moon during the greatest eclipse for a geocentric observer.
@@ 第41行： / 第53行： @@
 (d) (15 points) Determine the latitude and the longitude of the point where the axis of the Moon's shadow cone crosses the surface of the Earth during the greatest eclipse.
-===== Part II: Duration of the Totality (50 points) =====
+=== Part II: Duration of the Totality (50 points) ===
 Precisely determining the duration of totality of a solar eclipse involves complex calculations that would be beyond the scope of this problem. However, it is possible to obtain a reasonable approximation for this value using the two following assumptions:
 - The size of the umbra on the surface of the Earth remains roughly constant throughout totality at a given location.
 - The velocity of the umbra on the surface of the Earth remains roughly constant throughout totality at a given location.
@@ 第57行： / 第70行： @@
 (h) (14 points) Using System II, determine the velocity vector of the Moon during the greatest eclipse. Note that the intersection between the Celestial Equator and the lunar orbit that is closer to the position of the eclipse has a right ascension of \(23^{h}07^{m}59.2^s\).
-(i) (10 points) Write the velocity vector of the Moon in System III. Note that in System I, the azimuthal angle difference between the positions of the origins \(O_{\text{II}}\) and \(O_{\text{III}}\) is negligible, so you should only take into account the difference in the polar angles.
+(i) (10 points) Write the velocity vector of the Moon in System III. Note that in System I, the azimuthal angle difference between the positions of the origins OII and OIII is negligible, so you should only take into account the difference in the polar angles.
 (j) (6 points) Calculate the speed of the centre of the umbra along the surface of the Earth at the instant of the greatest eclipse.
 (k) (3 points) Estimate the duration of the totality of the eclipse at the location with the coordinates found in part (d).
 ==中文翻译==

@@ 第9行： / 第9行： @@
 Spherical Coordinates:
-{| class="wikitable"
 |    | Centre of the Sun    | Centre of the Moon    |
 |---|---|---|
 | Radial Distance (\(r\))   | \(1.516 \times 10^{11}\) m    | \(3.589 \times 10^8\) m    |
 | Polar Angle (\(\theta\))   | \(68^{\circ}29'44.1''\)    | \(68^{\circ}47'41.6''\)    |
-| Azimuthal Angle (\(\varphi\)) | \(-1^h11^{m}28.2^s\)    | \(-1^h11^{m}22.9^s\)    |}
+| Azimuthal Angle (\(\varphi\)) | \(-1^h11^{m}28.2^s\)    | \(-1^h11^{m}22.9^s\)    |
 Cartesian Coordinates:
-{| class="wikitable"|    | Centre of the Sun    | Centre of the Moon    |
+|    | Centre of the Sun    | Centre of the Moon    |
 |---|---|---|
 | \(x\)    | \(1.342 \times 10^{11}\) m    | \(3.185 \times 10^8\) m    |
 | \(y\)    | \(-4.327 \times 10^{10}\) m    | \(-1.025 \times 10^8\) m    |
-| \(z\)    | \(5.557 \times 10^{10}\) m    | \(1.298 \times 10^8\) m    |}
+| \(z\)    | \(5.557 \times 10^{10}\) m    | \(1.298 \times 10^8\) m    |
 For this problem, assume that the Earth is a perfect sphere.

@@ 第9行： / 第9行： @@
 Spherical Coordinates:
+{| class="wikitable"
 |    | Centre of the Sun    | Centre of the Moon    |
 |---|---|---|
 | Radial Distance (\(r\))   | \(1.516 \times 10^{11}\) m    | \(3.589 \times 10^8\) m    |
 | Polar Angle (\(\theta\))   | \(68^{\circ}29'44.1''\)    | \(68^{\circ}47'41.6''\)    |
-| Azimuthal Angle (\(\varphi\)) | \(-1^h11^{m}28.2^s\)    | \(-1^h11^{m}22.9^s\)    |
+| Azimuthal Angle (\(\varphi\)) | \(-1^h11^{m}28.2^s\)    | \(-1^h11^{m}22.9^s\)    |}
 Cartesian Coordinates:
-|    | Centre of the Sun    | Centre of the Moon    |
+{| class="wikitable"|    | Centre of the Sun    | Centre of the Moon    |
 |---|---|---|
 | \(x\)    | \(1.342 \times 10^{11}\) m    | \(3.185 \times 10^8\) m    |
 | \(y\)    | \(-4.327 \times 10^{10}\) m    | \(-1.025 \times 10^8\) m    |
-| \(z\)    | \(5.557 \times 10^{10}\) m    | \(1.298 \times 10^8\) m    |
+| \(z\)    | \(5.557 \times 10^{10}\) m    | \(1.298 \times 10^8\) m    |}
 For this problem, assume that the Earth is a perfect sphere.

@@ 第155行： / 第155行： @@
 ==官方解答==
-=== Part I: 地理坐标 ===
+=== 第一部分：地理坐标解答 ===
-(b) 太阳和月球的赤经（格林尼治恒星时为$5^h32^m35.5^s$）：
+(a) 赤纬计算：
-(c) 月球本影轴方向单位向量：
+$$ \delta = 90^\circ - \theta $$
-(d) 本影轴与地球表面交点坐标：
+太阳赤纬：
-通过方程$$|\vec{M} + k\vec{u}| = R_\oplus$$解得：
-$$$\phi = 4^\circ22'N$$$
+$$ \delta_\odot = 90^\circ - 68^\circ29'44.1'' = 21^\circ30'15.9'' $$
-$$$\lambda = 16^\circ42'W$$$
 (e) 本影半径估算：
-$$$r_{umbra} = 1.196 \times 10^5 \, \text{m}$$$
 (f) 地球自转线速度：
-(g) 月球轨道速度（忽略半长轴变化）：
+$$ v_{\text{rot}} = \frac{2\pi R_\oplus \cos\phi}{T_{\text{sidereal}}} = 463.7 \text{ m/s} $$
-$$$v_{Moon} = 1088 \, \text{m/s}$$$
-(h) 系统II中月球速度向量：
+$$ \vec{v}_{\text{III}} = \begin{bmatrix} v_x \\ v_y\cos\Delta\theta \\ -v_y\sin\Delta\theta \end{bmatrix} = \begin{bmatrix} 1085 \\ 77.76 \\ -23.54 \end{bmatrix} \text{ m/s} $$
-(i) 系统III中月球速度向量：
+(j) 本影移动速率合成：
-(j) 本影中心地表移动速度：
+$$ \vec{v}_{\text{umbra}} = \sqrt{(v_x - v_{\text{rot}})^2 + v_y^2} = 626.3 \text{ m/s} $$
 (k) 全食持续时间：
-$$$\Delta t = \frac{2r_{umbra}}{v_{umbra}} = 6\,\text{分钟}\,21.4\,\text{秒}$$$
 [[分类:日月食]]
 [[分类:球面三角]]
 [[分类:由ai生成]]

@@ 第80行： / 第80行： @@
 '''T10. 最大食分（75分）'''
-最大食分定义为月球本影锥轴最接近地球中心的时刻。本问题以1919年5月29日日食为例探讨该现象的几何学，此次日食因首次通过观测验证广义相对论而具有重大历史意义。科学考察队之一在巴西索布拉尔市观测了此次日食。
+最大食分定义为月球本影锥轴最接近地球中心的时刻。本题将探讨这一现象的几何原理，并以1919年5月29日日全食为例（该次日食具有重大历史意义，是天文学家首次通过观测验证广义相对论的契机）。其中一支科学考察队就是在巴西索布拉尔市进行观测的。
-下表展示了最大食分时刻太阳和月球的球面坐标与笛卡尔坐标。坐标系为右手系，原点在地球中心，$x$轴正向指向格林尼治子午线，$z$轴正向指向北极点。下文中称此坐标系为**系统I**。
+以下两个表格分别给出了最大食分时刻太阳和月球的球面坐标与笛卡尔坐标。该坐标系为右手坐标系，原点在地球中心，x轴正方向指向格林尼治子午线，z轴正方向指向北极点。在下文中称为"系统I"。
 {| class="wikitable"
 |-
 ! !! 太阳中心 !! 月球中心
 |-
-! 径向距离（$r$）
+| 径向距离 (\(r\)) || \(1.516 \times 10^{11}\) m || \(3.589 \times 10^8\) m
 |-
-! 极角（$\theta$）
+| 极角 (\(\theta\)) || \(68^{\circ}29'44.1''\) || \(68^{\circ}47'41.6''\)
 |-
-! 方位角（$\varphi$）
+| 方位角 (\(\varphi\)) || \(-1^h11^{m}28.2^s\) || \(-1^h11^{m}22.9^s\)
 |}
-'''笛卡尔坐标系：'''
+'''笛卡尔坐标：'''
 {| class="wikitable"
 |-
 ! !! 太阳中心 !! 月球中心
 |-
-! $x$
+| \(x\) || \(1.342 \times 10^{11}\) m || \(3.185 \times 10^8\) m
 |-
-! $y$
+| \(y\) || \(-4.327 \times 10^{10}\) m || \(-1.025 \times 10^8\) m
 |-
-! $z$
+| \(z\) || \(5.557 \times 10^{10}\) m || \(1.298 \times 10^8\) m
 |}
 ==官方解答==