Once the intersection points between the initial and transfer ellipse are found, the points are converted from x and y rectangular coordinates to radius, r, and true anomaly, θ. Since the origin of the rectangular coordinate frame is at the focus, the radius of the intersection point is given by

Because the initial and transfer ellipses share a common focus, this radius will be the same for each ellipse. The procedure for finding true anomaly is slightly more involved. First, the direction of the satellite is assumed to be orbiting in a clockwise direction about the focus. From the geometry shown in Figure 7.2, it is seen that the true anomaly (θ, in radians) is given by the following equation:

where α is a reference angle from an arbitrary x-axis and is given by

Figure 7.2  Representation of the true anomaly for a given point of intersection.

Once an intersection point of the two elliptical orbits is found in terms of r and θ, the velocities at both the initial and transfer ellipse () are found at this point using

The flight path angle, , for a given velocity is determined by

where the normal velocity is given by the equation,

The flight-path angle is determined for both the initial and transfer orbits. Since clockwise orbital motion is assumed, γ will be positive for 0° < θ < 180 ° and negative for 180° < θ < 360°.

The ΔV required for transfer at a particular intersection point is found using the law of cosines for the velocities at that point on the initial and transfer ellipses and the angle between these two velocities. This angle is represented in Figure 7.3. As can be seen, the angle between and is where and have opposite signs. The case where and have the same sign is shown in Figure 7.4. This figure illustrates that when and have the same sign, the angle between and is once again the difference of the two flight path angles. With this mind, the change in velocity required to maneuver from the initial orbit to the transfer orbit is

Because

the difference of the two flight path angles may be specified in either order, or , and the result will be the same.

Figure 7.3  Velocities and flight path angles for two orbits at an intersection point.

Figure 7.4  Flight path angles with the same sign.

The above process explains how to determine the ΔV required to maneuver a spacecraft from the initial to the transfer orbit. The process of determining the required ΔV must be performed for each intersection point of the initial and transfer orbit, and for each intersection point of the transfer and final orbit. Then, the minimum ΔV required to go from the initial to the transfer orbit is added to the minimum ΔV required to go from the transfer to the final orbit, resulting in the minimum total ΔV required to transfer from the initial to the final orbit. The fitness function is then found using a modification of equation (7.1), shown below in equation (7.13). For the fitness function, a constant value of 10 TU/DU is used because ΔV is usually of the order of 0 to 5DU/TU. (The canonical units of distance units (DU) and time units (TU) are used throughout this chapter). Making the constant value well above the expected range is important to avoid any negative fitness function values that may arise. Finally, the fitness function is multiplied by a factor (1,000) to give the GA a more substantial number to evaluate. The fitness function is

The GA calculates this fitness function for many populations in order to systematically search for the parameters of e, p, and ω that lead to the optimum fitness function value, thus minimizing total ΔV.