2018年IAO理论低年组第4题-科伦坡地球静止轨道卫星

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原文题目

Colombo. Geostationary satellite

4.1. At what minimun zenith distance can a geostationary satellite be observed from Colombo? Suppose that such a satellite is observed as a 2^m star in the night sky.

4.2. How long during a day(24^h) can we see this satellite with the naked eye (in a clear sky)?

4.3. Estimate teh size of the satellite, considering it a polished metallic sphere.

中文题目

科伦坡 地球静止卫星

4.1 从科伦坡可以观测到的地球静止卫星的天顶距最小为多少？ 假设这样的卫星与夜空中的2等星光度相等。

4.2 在一天（24小时）中我们可以用肉眼在晴朗的天空中看到这颗卫星多长时间？

4.3 估计卫星的大小，考虑它是抛光金属球（反射率为100%）。

英文解答

Geostationary satellite

4.1.Geostationary satellites are located above the same equator point. Terefore, each actual stellite is always visible at one point for a stationary observer on Earth.

It should be noted that the geostationary satellite should not be confused with a geosynchronous one, the orbit with a nonzero eccentricity or slight inclination with respect to the equator. Such a satellite in the sky describes loops and eights for observers on the Earth.

A geostationary satellite has a circular queatorial orbit with diameter R under the conditions:

$$\omega^{2}R_{st}\ =\ GM/R_{st}$$,

$$R_{st}^3\ =\ GM/\omega^2\ =\ GMT^2/4\pi^2$$,

where G is the gravitation constant, M is the mass of the Earth, T is the period of the Earth's revolution around its axis (23^h56^m04^s). The calculations give the result

$$R_{st}\ =\ 42\ 160\ km$$.

It is obvious that of all the possible geotaionary satellites, the smallest zenitn distance has a satellite located on the meridian of the observer. From the drawing we see that the zenith distance is z = φ + Ψ, where φ is the latitude,and Ψ is the angle at which the "equatorial pane - Colombo" segment is visble from the satellite. We can use the latitude os the plane in Colombo, which is indicated as"seafront in the center", 06°54′).

$$tan\psi\ =\ rsin\psi/(R-rcos\psi)$$,

The calculations give

$$\psi\ \approx\ 1°14′$$.

$$z\ \approx\ 08°08′$$.

4.2. This satellite can be observed all the dark time of day from the moment when 2^m stars become visible (apporximately the end of evening civil teilight) to the corresponding moment in the morning, except the possible period when the satellite falls into the shadow of the Earth. the indicated "dark time of day" is 11^h 12^m (12^h - 2×24 ^m) in average, and can continue in Colombo from 11^h00^m in summer to 11^h24^m in winter and in summer the satellite is never eclipsed by the Earth (and therefore visible all night), but in periods close to the equinoxes, such eclipses in the middle of the night are regular. In maximum they last approximately the time T = 24^h × 0.9·D_E / 2πR_st (we use here just 24^h, since the movement is aynodic, D_E is the diameter of the Earth, anf the coefficient 0.9 appears due to the narrowing of the earth shadow cone.

$$T_{max}\ =\ 62\ min$$.

This, in periods close to the equinoxes, when the "dark time of day" is 11^h12^m, the duration of visibility can be reduced by 62 minutes, that is, to 10^h20^m. Thus, the answer to the second question of the problem is: from 10^h20^m to 11^h24^m.

4.3. Let us compare the light fluxes coming to the observer (O) from the satellite (s) and from the full Moon (L).

If

W = light flux from the Sun near Earth,

a_i = albedo of the observed objects,

S_i = πr_i² = πd_i²/4, where r_i and d_i are the radius and the diameter of the observed objects respectively,

R_i = distances from the observed objects to the observer:

Flux from Moon to boserver: l_L = W·a_L·S_L/(2πR_LO²)

中文解答

4.1。地球静止卫星的星下点轨迹为赤道上的固定一点。因此，对于地球上的静止观察者来说，每个实际的司太立都总是在一个点可见。

应该指出的是，地球静止卫星不应与地球同步卫星相混淆，这种卫星的轨道具有非零偏心或相对于赤道略微倾斜。天空中的这种卫星描述了地球上观察者的环路和八个卫星。

地球静止卫星在以下条件下具有直径为R的圆形赤道轨道：

ω2Rst= GM /Rstω2Rst= GM / Rst，

R3st = GM /ω2= ​​GMT2 /4π2Rst3= GM /ω2= ​​GMT2 /4π2，

其中G是引力常数，M是地球的质量，T是地球围绕其轴旋转的周期（23h56m04s）。计算给出了结果

Rst = 42 160 kmRst = 42 160 km。

很明显，在所有可能的地质卫星中，最小的zenitn距离具有位于观察者子午线上的卫星。从图中我们看到天顶距离是z =φ+Ψ，其中φ是纬度，Ψ是“赤道窗格 - 科伦坡”段从卫星可见的角度。我们可以使用科伦坡的飞机纬度，它被称为“中心的海滨”，06°54'）。

tanψ=rsinψ/（R-rcosψ）tanψ=rsinψ/（R-rcosψ），

计算给出

ψ≈1°14'ψ≈1°14'。

z≈08°08'z≈08°08'。

4.2。这颗卫星可以在从2米星可见的时刻（大约是晚上民间的暮光之城）到早晨的相应时刻的所有黑暗时间观测，除了卫星落入地球阴影的可能时期。指示的“黑暗时间”平均为11h 12m（12h - 2×24m），并且可以在夏天从11h00m到冬季11h24m继续在科伦坡，夏天卫星永远不会被地球黯然失色（因此可见整夜），但在接近昼夜平分点的时期，半夜的这种日食是正常的。最大值约为时间T = 24h×0.9·DE /2πRst（我们这里使用的时间仅为24h，因为运动是aynodic，DE是地球的直径，因为地球阴影变窄，系数0.9出现锥体。

Tmax = 62minTmax = 62min。

在接近昼夜平分点的时期，当“黑暗时间”为11h12m时，能见度的持续时间可减少62分钟，即10h20m。因此，问题的第二个问题的答案是：从10h20m到11h24m。

4.3。让我们比较来自卫星和满月（L）的观察者（O）的光通量。

如果

W =来自地球附近太阳的光通量，

ai =被观察物体的反照率，

Si =πri2=πdi2/ 4，其中ri和di分别是被观察物体的半径和直径，

Ri =从观察对象到观察者的距离：

从月球到支架的通量：lL = W·aL·SL /（2πRLO2）