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.

解答

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

$$\psi\ \approx\ 1°14′$$.

$$z\ \approx\ 08°08′$$.