Orbital node

Abstract: A J2 dynamic model of satellite relative motion in the form of differential equations with reference to the local vertical local horizontal (LVLH) coordinate is basic and critical to the study of satellite formation flying. Many versions [1–7] of J2 dynamic equations have been published. Thework [1] byKechichian presents an exact nonlinear relative model taking into account both J2 perturbation and air drag. Kechichian applied techniques of Newtonian mechanics and vector calculus to derive the relative dynamics. However, his model is very complex and does not explicitly include some critical components of the J2 acceleration. The calculation of these J2 acceleration components is by a tedious algorithm. The model’s complexity hampers its application in control designs. Other published dynamic models [2–7] were developed assuming the reference orbit as being unperturbed Keplerian motion; thus, modeling errors are introduced. In this Note, the satellite relative dynamics is studied based on the perturbed reference orbit, which is accurately described by a set of differential equations. We express the reference orbit in terms of reference satellite variables (RSV) and obtain a simpler representation than that by Kechichian. Furthermore, we use Lagrangian mechanics to derive the satellite relative dynamics, which is different from Kechichian’s approach. As a result, we establish an exact J2 nonlinear relative model independent of the right ascension of the ascending node. The satellite relativemotion is explicitly expressed in terms of very few physical parameters and a set of simple first-order differential equations. This simplicity is due to the fact that both the spherical and the J2 accelerations are independent of the change of the right ascension of the ascending node. II. J2 Reference Satellite Dynamics in a Rotating Frame

Abstract: The effect of orbit plane precession is used to place a plurality of satellites into one or more desired orbit planes. The satellites are distributed within each desired orbit plane in a selected configuration. The satellites are transported into orbit on one or more frame structures referred to as "pallets". When more than one pallet is used, they are placed on top of each other in a "stack". After the stack of the pallets has been launched into an initial, elliptical orbit, the pallets are separated sequentially from the stack at selected time intervals. Thrust is applied to transfer a first pallet from the initial orbit to a first, circular orbit, wherein the initial and first orbits are in planes that process at different predetermined initial and first rates, respectively. After waiting for a predetermined time while the initial orbit plane and the first orbit plane precess with respect to each other, thrust is applied to the next pallet to transfer it into a next, circular orbit in a next orbit plane, wherein the precession rate of the next orbit plane also is different from the initial precession rate of the initial orbit plane. The foregoing step is repeated until the satellites on the respective pallets have been sequentially deployed into the desired orbit planes. The satellites on each pallet are then separated from the pallet simultaneously, but at different rates to achieve separation among the satellites within each orbit.

Abstract: In 1693 G. D. Cassini published the three following empirical laws on the Moon’s rotational motion: (1) The Moon rotates uniformly about its polar axis with a rotational period equal to the mean sidereal period of its orbit about the earth. (2) The inclination of the Moon’s equator to the ecliptic is a constant angle approximately 1°5. (3) The ascending node of the lunar orbit on the ecliptic coincides with the descending node of the lunar equator on the ecliptic.

Abstract: Presented is an analytic method to establish J2 invariant relative orbits, that is, relative orbit motions that do not drift apart. Working with mean orbit elements, the secular relative drift of the longitude of the ascending node and the argument of latitude between two neighboring orbits are set equal. Two first order conditions constrain the differences between the chief and deputy momenta elements (semi-major axis, eccentricity and inclination angle), while the other three angular differences (ascending node, argument of perigee and mean anomaly), can be chosen at will. Several challenges in designing such relative orbits are discussed. For near polar orbits or near circular orbits, enforcing the equal nodal rate condition may result in impractically large relative orbits if a difference in inclination angle is prescribed. In the latter case, compensating for a difference in inclination angle becomes exceedingly difficult as the eccentricity approaches zero. The third issue discussed is the relative argument of perigee and mean anomaly drift. While this drift has little or no effect on the relative orbit geometry for small or near-zero eccentricities, for larger eccentricities it causes the relative orbit to enlarge and contract over time. A simple control solution to this issue is presented. Further, convenient expressions are presented which allow for quick annual fuel budget estimations. For given initial orbit element differences, these formulas estimate what Δv is required to compensate for the J2 induced relative drift.