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Homework Help: Interruption of solar observation on a sun-synchronous orbit

  1. Nov 9, 2013 #1
    1. The problem statement, all variables and given/known data
    Analyse the interruption of solar observation of a satellite on a sun-synchronous orbit (altitude 900km above Earth's surface). The orbit crosses the equatorial plane at the day/night border and the orbital plane should follow this border as close as possible.
    a) Draw the possible orbits at the start of summer and determine the angle between Earth's rotational axis and orbit plane and the angle between Sun direction and equator plane.
    b) Does the satellite enter the shadow zone during passage of the pole? If yes, how long is the eclipsed period.


    2. Relevant equations
    β is the angle between Earth-Sun vector s and the orbit plane. It can be derived using the normal n of the orbit plane: [itex]\beta = \sin^-1(s \cdot n)[/itex]
    The Earth's central angular radius [itex]\beta^\ast[/itex] is defined as:
    [tex]\beta^\ast = \sin^{-1}(\frac{R_E}{h+R_E})[/tex]
    The eclipsed fraction of the orbit [itex]F_e[/itex] depending on β is defined as:
    [tex]F_e=\frac{\cos^{-1}(\frac{\cos \beta^ast}{\cos \beta})}{180^\circ}=\frac{1}{180^\circ}\cdot \cos^{-1}(\frac{\sqrt{h^2 + 2 R_E h}}{(R_E + h) \cos \beta})[/tex]

    3. The attempt at a solution
    Regarding a): As you can see I already drew the possible orbits. At first I assumed that we only have one possible orbit. But I think we do have two orbits, which only differ in the direction in which the satellite travels (for better understanding I tilted the red and green orbit, although they are not tilted).

    Regarding the angles: Intuitively I would assume that both angles (Sun and equator plane, and Earth's rotational axis and orbit plane) equal the Earth's tilt of 23.5 degree. However this is just a guess and I do not have any formula for it. Is there a way to compute it?

    Regarding b): I would like to use the formulas I provided here for [itex]\beta^\ast[/itex] and [itex]F_E[/itex]. However I am not sure about the values [itex]h[/itex] and [itex]\beta[/itex]. Is [itex]h[/itex] the height of the satellite meaning the 900km or has it to be 7278 km (taking Earth radius into account?).
    Furthermore I would guess [itex]\beta = 23.5^\circ[/itex] (see "Regarding the angles"). However I am not sure about β value, especially because I have no clue how to identify the Sun-vector or the Earth's orbital normal vector (as these vectors can be used to compute beta) ...

    Thank you very much for your help!
  2. jcsd
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