2024年IOAA理论第11题-地面轨迹
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目录
英文题目
T11. Ground Tracks (75 points)
The projection of a satellite's orbit onto the Earth's surface is called its ground track. At a given instant, one can imagine a radial line drawn outward from the centre of the Earth to the satellite. The intersection between the Earth's spherical surface and this radial line is a point on the ground track.
The location of this point is specified by its geocentric latitude and longitude. The ground track is then essentially the figure traced by this point as the satellite moves around the Earth.
Part I: Sun-Synchronous Orbits (25 points)
It is particularly interesting to analyse the ground track of a so-called Sun-Synchronous orbit. This is a nearly polar orbit around a planet where the satellite passes over any given point on the planet's surface at the same mean local solar time. This property is especially interesting for satellite imaging, ensuring similar illumination conditions over different days.
The figure below shows the ground track of a satellite in a Sun-Synchronous orbit. Its inclination angle \(i\) - the angle between the satellite orbital plane and the Earth's equatorial plane - falls within the range \(90^\circ < i < 180^\circ\). The graph depicts five complete orbits of the satellite.
Figure 2 - Ground Track for five orbital periods of the satellite
For the questions in Part I, assume that the Earth's orbit around the Sun is circular.
(a) (3 points) Determine the nodal precession rate for the orbit in rad/s.
(b) (8 points) Based on the ground track shown in Figure 2, determine the inclination of the satellite's orbit (in degrees) and estimate its orbital period (in minutes). Consider that the orbital period of the satellite is shorter than one sidereal day.
(c) (2 points) Calculate the semi-major axis \(a\) of the orbit in km.
(d) (1 point) Determine the number of orbits completed by the satellite until it returns to the same position on Earth.
(e) (11 points) As seen in Figure 2, the ground track crosses the Brazilian city of Maceió \((\phi,\lambda)=(9.7^\circ S;35.7^\circ W)\) and also Chorzów \((\phi,\lambda)=(50.3^\circ N;19.0^\circ E)\), in Poland. Knowing that the ground track crosses Maceió at noon (local time), determine the local time that the satellite crosses Chorzów. Hint: specifically for this task, you may neglect the effects of nodal precession.
Part II: Tundra orbits (50 points)
A Tundra orbit is a type of geosynchronous elliptical orbit characterised by a high inclination. The apogee is positioned over a specific geographic region, allowing for prolonged visibility and coverage over that area. This orbit ensures that a satellite spends the majority of its orbital period over the northern - or southern - hemisphere, making it particularly useful for communications and weather observation over high-latitude regions.
The image below represents the ground track of a satellite in a Tundra orbit with an argument of perigee equal to \(270^\circ\). The satellite orbits the Earth in the same direction as its rotation. For the following items, you can ignore the Earth's oblateness.
Figure 3 - Tundra Orbit Ground Track for one orbital period of the satellite
(f) (4 points) Based on the graph above, give the inclination of the satellite's orbit \(i\) (in degrees), its orbital period \(T\) (in minutes), and its semi-major axis \(a\) (in km).
(g) (12 points) Show that the time a satellite spends in the northern hemisphere is given by \[ T'=\left(\frac{1}{2}+\frac{\sin^{-1}(e)}{\pi}+\frac{e}{\pi}\cdot\sqrt{1-e^{2}}\right)T \] where \(e\) is the eccentricity of the orbit and \(T\) is its orbital period.
(h) (10 points) Estimate numerically the eccentricity \(e\) of its orbit. You can consider that the eccentricity is so small that \(\sin(e)\approx e\) and \(e^{2}\ll1\).
(i) (18 points) From the ground track, we can observe that the satellite exhibits retrograde motion in both its northern and southern hemisphere trajectories. Find the true anomaly (in degrees) of the satellite at the beginning and end of its retrograde motion in the southern hemisphere.
(j) (6 points) It is also noticeable that the ground track of a Tundra orbit has the shape of a figure-8, similar to an analemma, so that the satellite passes over the same point on Earth in a single orbit. Calculate the minimum eccentricity the orbit would need to have for this property to cease occurring. Use the same orbital inclination as the orbit in Figure 3.
中文翻译
T11. 星下点轨迹（75分）
卫星轨道在地球表面的投影称为地行迹。在某一时刻，可以想象一条从地心向外延伸到卫星的径向线。地球表面与这条径向线的交点就是地行迹上的一个点。
该点的位置由其地理经纬度指定。地行迹本质上是卫星绕地球运行时，该点所绘出的轨迹图形。
第一部分：太阳同步轨道（25分）
分析所谓的太阳同步轨道的地行迹特别有趣。太阳同步轨道是围绕行星运行的近极轨道，卫星在相同的地方平太阳时经过行星表面的任何给定点。此特性对于卫星成像特别有用，可确保不同日子的照明条件相似。
下图显示了一颗太阳同步轨道卫星的地行迹。其倾角\(i\)（卫星轨道平面与地球赤道平面之间的角度）满足\(90^\circ < i < 180^\circ\)。图 中画出了该卫星的五个完整轨道周期。
图2 - 卫星五个轨道周期的地行迹
回答第一部分的问题时，假设地球绕太阳的轨道为圆形。
(a)（3分）求轨道的交点进动率，单位为rad/s。
(b)（8分）根据图2的地面轨迹，求卫星轨道倾角（单位为度)，并估算其轨道周期（单位为分钟)。考虑到卫星的轨道周期短于一个恒星日。
(c)（2分）计算轨道半长轴\(a\)（单位为km）。
(d)（1分）求卫星回到地球表面同一位置需要多少个完整的轨道周期。
(e)（11分）如图2所示，地面轨道穿过巴西城市Maceió，\((\phi,\lambda)=(9.7^\circ S;35.7^\circ W)\)，以及波兰城市Chorzów\((\phi,\lambda)=(50.3^\circ N;19.0^\circ E)\)。已知地行迹在正午（地方时）穿过Maceió，求卫星轨迹穿过Chorzów的地方时。提示：对于此问题，可以忽略交点进动的影响。
第二部分：苔原轨道（50分）
苔原轨道是一种地球同步椭圆轨道，其特点是倾角较大。远地点位于特定地理区域的上方，可延长卫星在该区域的观测时间并扩大覆盖范围。该轨道可确保卫星在大部分轨道周期内位于北半球或南半球，特别适合高纬度地区的通信和气象观测。
原图表示苔原轨道上一颗卫星在一个周期内的地行迹，其近地点幅角为270°。卫星的轨道运动方向与地球自转方向相同。对于以下问题，可以忽略地球扁率的影响。
图3 - 单圈轨道周期的苔原轨道的地行迹
(f)（4分）基于上述地行迹，给出卫星轨道倾角\(i\)（单位为度）、轨道周期\(T\)（单位为分钟），以及轨道半长径\(a\)（单位为km)。
(g)（12分）推导卫星在北半球的停留时间表达式 \[ T'=\left(\frac{1}{2}+\frac{\sin^{-1}(e)}{\pi}+\frac{e}{\pi}\cdot\sqrt{1-e^{2}}\right)T \] 其中\(e\)是轨道偏心率，\(T\)是轨道周期。
(h)（10分）估计该轨道的偏心率\(e\)。可以认为偏心率很小，\(\sin(e)\approx e\)且\(e^{2}\ll1\)。
(i)（18分）从地面轨迹上可以观察到，卫星在北半球和南半球轨道上都呈现逆行运动。求卫星在南半球逆行运动开始和结束时的真近点角（单位为度）。
(j)（6分）苔原轨道的地行迹呈类似日行迹的“8”字形，使卫星在一个轨道周期内经过地球上的同一点。求使这一性质刚好不再出现所需的最小轨道偏心率。使用图3中相同的轨道倾角。
