Interruption of solar observation on a sun-synchronous orbit

[Paul Gray]

Homework Statement

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.

Homework 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: $\beta = \sin^-1(s \cdot n)$
The Earth's central angular radius $\beta^\ast$ is defined as:
$$\beta^\ast = \sin^{-1}(\frac{R_E}{h+R_E})$$
The eclipsed fraction of the orbit $F_e$ depending on β is defined as:
$$F_e=\frac{\cos^{-1}(\frac{\cos \beta^ast}{\cos \beta})}{180^\circ}=\frac{1}{180^\circ}\cdot \cos^{-1}(\frac{\sqrt{h^2 + 2 R_E h}}{(R_E + h) \cos \beta})$$

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 $\beta^\ast$ and $F_E$. However I am not sure about the values $h$ and $\beta$. Is $h$ the height of the satellite meaning the 900km or has it to be 7278 km (taking Earth radius into account?).
Furthermore I would guess $\beta = 23.5^\circ$ (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!

 

[anathema]

Thank you for your interesting post. I am happy to provide some insights and suggestions for your analysis.

Regarding a): Your assumption about having two possible orbits is correct. The satellite can either travel in the direction of Earth's rotation (prograde orbit) or against Earth's rotation (retrograde orbit). The tilt of the orbit plane in both cases will be the same, but the direction of travel will be different.

To determine the angles, you can use the formula β = sin^-1(s · n). Here, s is the Earth-Sun vector, which is always pointing towards the Sun, and n is the normal vector of the orbit plane. Since the orbit crosses the equatorial plane at the day/night border, the normal vector will be perpendicular to the equatorial plane. Therefore, the angle between the Sun direction and the equator plane will be the same as the angle between the Sun direction and the normal vector, which is the same as the Earth's tilt of 23.5 degrees.

Similarly, the angle between Earth's rotational axis and the orbit plane will also be 23.5 degrees, as the orbit plane is tilted with respect to the equator plane.

Regarding b): The value of h in the formulas you provided represents the height of the satellite above Earth's surface, so it should be 900 km in this case. The value of β will also be 23.5 degrees, as mentioned above.

To determine the eclipsed period, you can use the formula Fe = (1/180) * cos^-1(√(h^2 + 2REh)/(RE + h) * cos β). This formula gives you the fraction of the orbit that is in the shadow zone, so you can multiply it by the orbital period to get the duration of the eclipsed period.

I hope this helps. Let me know if you have any further questions or need clarification. Good luck with your analysis!