Satellite Orbit and Ephemeris Determination using Inter Satellite Links

Inter Satellite LinksOrbit ComputationRadius:pr =1 + ε cos ϕEq. 4.1-82Ellipse parameter: p = a(1 − ε )Eq. 4.1-9T2πTime of flight: t − T0 = ⋅ ( M − εsin M)Eq. 4.1-10Eccentric anomaly:⎛ ε + cosϕ⎞E = arccos⎜⎟⎝1+ ε ⋅ cosϕ⎠Eq. 4.1-11Mean anomaly:M = E − ε ⋅sinEEq. 4.1-12Mean Motion:n =360T=2πTEq. 4.1-13εsinϕFlight path angle: tan γ =1+ εsinϕEq. 4.1-14with ϕ true anomalyM mean anomalyT orbital periodTo obtain three dimensional Cartesian co-ordinates, the ellipse parameters have to betransformed to Cartesian vector using the following expression:x OP⎧r cos ϕ⎫⎪ ⎪= ⎨r sin ϕ⎬⎪ ⎪⎩ 0 ⎭Eq. 4.1-15The index OP indicates a reference frame lying in the orbital plane with the x-axis coincidingwith the line of apsis, the z-axis normal to the orbit plane and the origin being the focus of theellipse.The transformation from the orbital plane frame to an inertial fixed frame (e.g. J2000) is doneapplying the following vector-matrix operation.R. Wolf Page 21