(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A satellite is launched into a circular sun-synchronous orbit at a height of 900km above Earth's surface. What is the implication on the orbit's inclination (in [itex]deg[/itex]) and on the change of the position of the right ascension of the ascending node per day.

2. Relevant equations

The change of the right ascension of the ascending node is defined as:

[tex]\Delta\Omega = - \frac{3\pi J_2 R^2_E}{a^2(1-\epsilon^2)^2}\cos i \text{ [rad/rev]}[/tex]

and the change of the argument of perigee is defined as

[tex]\Delta\omega = - \frac{3\pi J_2 R^2_E}{a^2(1-\epsilon^2)^2}(4-5 \sin^2 i) \text{ [rad/rev]}[/tex][tex]R_E=6378km\\

J_2=1082.7 \cdot 10^{-6}[/tex]

3. The attempt at a solution

In a previous task I already identified [itex]a = 7278 km[/itex].

Since we have a sun-synchronous orbit, the satellite-sun vector has to be constant and equals the earth-sun vector. Hence I assume I can calculate [itex]\Delta\Omega[/itex] as following. I'm using sidereal days and the information that the Earth performs a [itex]360^\circ[/itex] rotation during 1 year.

[tex]\frac{\Delta\Omega}{360^\circ} = \frac{23.9345h}{365d \cdot 23.9345h} \Rightarrow \Delta\Omega = 0.9863 \text{[deg/day]}[/tex]

So now the only unknown variable in the formula for the change of the right ascension of the ascending node is [itex]i[/itex]:

[tex]\Delta\Omega = - \frac{3\pi J_2 R^2_E}{a^2(1-\epsilon^2)^2}\cos i \\

\Rightarrow i = \cos^{-1}(\frac{a^2(1-\epsilon^2)^2}{- 3\pi J_2 R^2_E}\cdot \Delta\Omega) = \cos^{-1}(\frac{7278^2(1-0^2)^2}{- 3\pi \cdot 1082.7 \cdot 10^{-6} 6378^2}\cdot 0.9863) = \cos^{-1}(-103.251)[/tex]

As you can see this produces aMATH ERRORon my calculator :). Unfortunately I am not able to identify where I went wrong :). Can you please help me :)?

Thanks in regard!

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# Sun-synchronous orbit: Implication on the orbit's inclination

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