# Obtaining orbital parameters of a closed orbit given the characteristics of the arrival hyperbolic orbit

According to this paper (page 8) by ESA, given the $$v_{inf}$$, the right-ascension and the declination of the arrival hyperbolic asymptote, it is possbile to find a loci of pericenters of different closed orbits (with varying inclination, argument of pericenter and RAAN) by varying the B-angle.

I would like to know how to determine how the closed orbits' orbital parameters (in this case, the inclination, the argument of pericenter and the RAAN) change by varying the B-angle.

I don't have a derivation readily available, but here are some hints on solving this.

1. The B-Plane vectors are defined by cross products of the eccentricity and orbital momentum vectors: https://nyxspace.com/MathSpec/celestial/orbital_elements/#b-plane-b_plane.
2. The inclination is the arc cosine of the Z component of the orbital momentum vector over the magnitude of the momentum: https://nyxspace.com/MathSpec/celestial/orbital_elements/#inclination-inc .
3. The argument of periapsis is the arccos of the normal to the orbital momentum and the eccentricity vector (normalized): https://nyxspace.com/MathSpec/celestial/orbital_elements/#argument-of-periapsis-aop
4. The RAAN is the arccos of the normal to the orbital momentum: https://nyxspace.com/MathSpec/celestial/orbital_elements/#right-ascension-of-the-ascending-node-raan .
5. Finally, the B vector angle is the arctan of the B plane parameters: https://nyxspace.com/MathSpec/celestial/orbital_elements/#b-vector-angle-angle .

With all of that (and the other orbital element formulations as described in the sources above), you should be able to derive the variation in any parameters wrt the B Plane angle.

• How do I determine the momentum vector (relative to the target body) or the position and velocity vectors when I solve Lambert’s problem? – Miguel Apr 21 at 17:27
• @Miguel, what information do you currently have on your trajectory? You may computer the momentum vector relative to the target body by converting the position and velocity of your spacecraft into the body-centered inertial frame (e.g. convert from an Earth centered frame to a Moon centered frame). – ChrisR Apr 21 at 19:45
• I only have the v-infinity arrival vector and the orbital parameters of the hyperbolic orbit (on a 2D plane). I also have the 2D orbital parameters of the elliptical orbit that I want to achieve. The problem is that if I'm using the patched conics method (Earth-to-Mars mission), then how do I convert the spacecraft's position vector if it's the same as the arrival planet's position vector. – Miguel Apr 21 at 20:48
• Is your v_inf vector also in 2D? If so, you can assume that everything is on the same plane, so you'd simply "extend" it to 3D by setting the Z component to zero. However, to convert to a body frame, you'll need to use the method from your problem as a real ephemeris won't do. – ChrisR Apr 21 at 21:38
• My arrival v_inf vector has 3 components and the way I calculated the declination and right-ascension of the arrival asymptote is by doing the same method I answered here. According to the paper that I attached to this question, knowing the v_inf magnitude, DAP and RAP it is possible to obtain different elliptical/circular orbits by changing the B-angle, but I still don't understand how I should obtain the parameters of these closed orbits. – Miguel Apr 22 at 20:59