Mission Design for the CubeSat OUFTI-1

CHAPTER 8We have now the vector ˆN s expressed into the orbit reference:⎧ ⎫ ⎧ ⎫⎨ x orb ⎬ ⎨ x ecl ⎬y orb⎩ ⎭ = R 4R 3 R 2 R 1 y ecl⎩ ⎭z orb z ecl(8.6)As the satellite is moving on the orbit plane, what we are interested in tocalculate the eclipses time is actually the projection of N s on the orbit plane.As indicated in figure 8.1, we calculate the angle β that the projection of N sgenerates with the x orb :( )Ns,yN s = atanN s,x(8.7)Figure 8.3: Sun rays direction projected on the orbit plane.So far, we know the eclipse’s central angle θ ∗ = 180 ◦ + β and therefore weknow the corresponding orbit radius.As the distance between earth and sun is much bigger than the earth’s radius,we can make the hypothesis that the lines determining the entrance and the exitfrom eclipses are tangent to the earth surface, as shown in figure 8.1. Hence,we have:¯θ out = 90 ◦ − acos(Rer out)¯θ in = 90 ◦ − acos(Rer in) (8.8)We can also exploit the relationship between radius and anomaly and wehave:Galli Stefania 76 University of Liège