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If there are two circular orbits, each with the same semi-major axis, and the two orbits only differ in inclination and RAAN; what is the intersection angle between the two? Example: two satellites in two such orbits, both with RAAN=0 and one with inc=0 and one with inc=90; these two satellites would have a strike angle of 90 degrees relative to their velocity vectors. But what is the formula for calculating the intersect angle for any given RAAN and inclination?

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  • $\begingroup$ Welcome into the Py.SE. Excuse me, but what are RAAN and inc? $\endgroup$
    – Sebastiano
    Sep 2, 2019 at 19:19
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    $\begingroup$ Right Ascension of the Ascending Node (RAAN) and inclination. Instead of RAAN, it can be longitude of the ascending node. The important point is that the orbits are circular but differ in those two elements. $\endgroup$
    – astrodyn
    Sep 2, 2019 at 19:48

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The angle at which the orbits intersect is the same as the angle between their planes, which is the same as the angle between their normal directions.

Thus the intersection angle $\alpha$ is given by

$$\cos\alpha=\hat{\mathbf{n}}_1\cdot\hat{\mathbf{n}}_2=\cos\theta_1\cos\theta_2+\sin\theta_1\sin\theta_2\cos(\phi_1-\phi_2)$$

where $\theta$ is the inclination, $\phi$ is the RAAN, and

$$\hat{\mathbf{n}}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$$

is a unit vector which is normal to the orbit.

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  • $\begingroup$ Thank you, G. Smith. That was exactly what I needed. $\endgroup$
    – astrodyn
    Sep 3, 2019 at 2:45

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