Space methods - Physical Geodesy - page 262

obtained by

⎡
⎣

⎤
⎦

⎡
⎣

⎤
⎦

X 0
Y 0
Z 0

cos v
sin v
0

= R r , (7-32)

where the matrix R is composed of three successive rotation matrices (see
Figs. 7.2 and 7.3) and is given by

R = R 3 {− Ω } R 1 {− i } R 3 {− ω }

⎡ ⎤

cos Ω cos ω − cos Ω sin ω sin Ω sin i

⎣ ⎦

− sin Ω sin ω cos i − sin Ω cos ω cos i

= sin Ω cos ω − sin Ω sin ω − cos Ω sin i , (7-33)

+cosΩsin ω cos i + cos Ω cos ω cos i

sin ω sin i cos ω sin i cos i

see Hofmann-Wellenhof et al. (2001: p. 43). The column vectors of the or-
thonormal matrix R are the axes of the orbital coordinate system represented
in the equatorial system X 0

i .
Substituting (7-33) into (7-32) and carrying out the multiplication (Mon-
tenbruck and Gill 2001: Eq. (2.51)) yields

X 0 = r [cos Ω cos( ω + v ) − sin Ω sin( ω + v )cos i ] ,

Y 0 = r [sin Ω cos( ω + v )+cosΩ sin( ω + v )cos i ] , (7-34)

Z 0 = r sin( ω + v )sin i,

where, according to (7-5),

e 2 )
1+ e cos v .

a (1 −

r = (7-35)

This expresses the rectangular coordinates of the satellite in terms of the
elements of its osculating orbit, the true anomaly v fixing its position as a
function of time.
Since the osculating ellipse does not remain constant, it is convenient
to use a fixed reference orbit - for instance, the osculating ellipse E 0 at
a certain instant t 0 , having the elements a 0 ,e 0 ,i 0 , Ω 0 ,ω 0 ,T 0 .Atalater
instant t , the orbital elements will have changed to a 0 +∆ t a, e 0 +∆ t e, i 0 +
∆ t i, Ω 0 +∆ t Ω ,ω 0 +∆ t ω, T 0 +∆ t T , which corresponds to an osculating
ellipse E t .