Satellite Orbit and Ephemeris Determination using Inter Satellite Links

Software DescriptionInter Satellite Links5.4 Co-ordinate TransformationIn an orbit simulation / orbit determination process a lot of information will be needed indifferent reference frames. For example, the satellites equations of motion are described in aninertial frame, while the coordinates of a tracking station or user will be given in an earthcentred-earth-fixedframe. Therefore it will be necessary to transform force, velocity andposition vectors from one frame to another.This is done using rotation matrices. To perform a complete transformation from inertial (ECIJ-2000) to earth-centred-earth-fixed (ECEF), one has to account for four different effects.• Precession• Nutation• Polar Motion• Sidereal TimeRECEFECI= R ⋅ R ⋅ R ⋅REq. 5.4-1PMSNPrNote that a matrix multiplication is non commutative, but orthogonal rotation matrices havethe following property1 T ECIR = R ⇒ RECEF=− REq. 5.4-2ECEF TECIi.e. transformation matrix for the backward transformation is simply obtained from thetransponed. The following equations are found in [HWL-94] or in the Astronomical Almanac.5.4.1 PrecessionThe transformation matrix accounting for precession is given byR P⎡cos z cos ϑcosζ⎢−sin z sin ζ⎢=sin z cos ϑcosζ⎢+cos z sin ζ⎢⎢ sin ϑcosζ⎣− cos z cos ϑsinζ− sin z cos ζ− sin z cos ϑsinζ+ cos z cos ζ− sin ϑsinζ− cos z sin ϑ⎤⎥⎥− sin z sin ϑ⎥⎥cos ϑ ⎥⎦where the necessary Euler angles are derived fromEq. 5.4-323ζ = ⋅ + ⋅ + ⋅Eq. 5.4-42306".21810".30188z = 2306".2181⋅T + 1".09468 ⋅ Tϑ = 2004".3109 ⋅ T − 0".42665 ⋅ TTT2+ 0".018203 ⋅ T20".017998T3− 0".041833 ⋅ TT is the time interval between the observation date and the J2000.0 standard epoch, expressedin Julian centuries. One Julian century has 36525 days. Note that the transformation anglesζ,ϑ,z are given in arc seconds. They have to be scaled radians prior to further use in equation5.4-3.3Page 84R. Wolf