## Orbit Maneuvering

At some point during the lifetime of most satellites, we must change one or more of the orbital elements. For example, we may need to transfer it from an initial parking orbit to the final mission orbit, rendezvous with or intercept another satellite, or conect the orbital elements to adjust for the perturbations discussed in the previous section. Most frequently, we must change the orbital altitude, plane, or both. To change the oibit of a satellite, we have to change the satellite's velocity vector in magnitude or direction using a thruster. Most propulsion systems operate for only a short time compared to the orbital period, so we can treat the maneuver as an impulsive change in the velocity while the position remains fixed. For this reason, any maneuver changing the orbit of a satellite must occur at a point where the old orbit intersects the new orbit If the two orbits do not intersect, we must use an intermediate orbit that intersects both. In this case, the total maneuver requires at least two propulsive burns.

In general, the change in the velocity vector to go from one orbit to another is given by

We can find the current and needed velocity vectors from the orbital elements, keeping in mind that the position vector does not change significantly during impulsive burns.

### 6.3.1 Coplanar Orbit Transfers

The most common type of in-plane maneuver changes the size and energy of the orbit, usually from a low-altitude parking orbit to a higher-altitude mission oibit such as a geosynchronous orbit. Because the initial and final orbits do not intersect (see Fig. 6-7), the maneuver requires a transfer orbit Figure 6-7 represents a Hohmann' Transfer Orbit In this case, the transfer orbit's ellipse is tangent to both the initial and final circular orbits at the transfer orbit's perigee and apogee, respectively. The orbits are tangential, so the velocity vectors are collinear at die intersection points, and the Hohmann Transfer represents the most fuel-efficient transfer between two circular, coplanar orbits. When transferring from a smaller orbit to a larger orbit, the propulsion system must apply velocity change in the direction of motion; when transferring from a larger orbit to a smaller, the velocity change is opposite to the direction of motion.

The total velocity change required for the transfer is the sum of the velocity changes at perigee and apogee of the transfer ellipse. Because the velocity vectors are collinear at these points, the velocity changes are just the differences in magnitudes of the velocities in each orbit. We can find these differences from the energy equation, if we know the size of each orbit. If we know the initial and final orbits (rA and rB), we calculate the semimajor axis of the transfer ellipse, am and the total velocity change

* Walter Hohmann, a German engineer and architect, wrote The Attainability of Celestial Bodies [1925], consisting of a mathematical discussion of the conditions for leaving and returning to Earth.

Final Orbit

Initial Orbit

Final Orbit

Initial Orbit

Hohmann Transfer

Fig. 6-7. Hohmann Transfer. The Hohmann Transfer ellipse provides orbit transfer between two circular, co-planar orbits.

Hohmann Transfer

Fig. 6-7. Hohmann Transfer. The Hohmann Transfer ellipse provides orbit transfer between two circular, co-planar orbits.

(the sum of the velocity changes required at points A and B) using the following algorithm. An example transferring from an initial circular orbit of6,367 km to a final circular orbit of42,160 km illustrates this technique.

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