Khamseh, H.B., Navabi, M.
Figure 1: Gaps in ground segment access to a satellite.
relationships to obtain position vector of the satellite. In this method, Lagrange planetary equations are employed with the second-degree gravitational potential function. Lagrange planetary equations can be found in references such as Capderou (2005) and are given by Eq. 1: ¯ ² ² ² ² ² ² ² ²² ° ² ² ² ² ² ² ² ² ²¹
da 1 Š yR š " 2 dt na ª yM º yR yR š 1 1 e2 Š 1 de " 2 ª º 2 e  dt na 1 e y\ yM  Š yR di 1 yR š " cos i ª 2 2 dt na 1 e sin i  y< y\ º Š yR š 1 d< " 2 2 dt na 1 e sin i ª yi º Š 1 e2 yR cos i yR š 1 d\ " dt na 2 1 e2 ª e ye sin i yi º 1 Š yR 1 e2 yR š dM " n 2 ª 2 a ya dt e ye º na 
`
(1)
b
In Eq. 1, a, e, i, <, \ , M is the Keplerian set of orbital elements namely semi-major axis, eccentricity, inclination, Right Ascension of Ascending Node (RAAN), argument of perigee and mean anomaly, respectively. Also n " R3 is a the mean motion, Âľ is Earth gravitational parameter and R is perturbing Geopotential. A second-degree perturbing Geopotential takes account of J2 effect i.e. dominant perturbation of LEO region and thus is employed in this Â&#x2022; cos 1 cos t < sin 1 sin t cos i study. If we only take account a(1 < e2 ) ofÂłthe secular variations of ď ˛ sin elegant 1 cos t +relationship cos 1 sin t cos i rS (t )we = may obtainÂł an orbital elements, 1 + e cos Â&#x17E; (t ) sin iperturbing sin t for the average second-degree ÂłÂ&#x2013;gravitational function, i.e. R J 2. The procedure to obtain R J 2 is given by Capderou (2005) and the result is given by Eq. 2: R J2 "
54
3 2
R Re2
a3 1 e
3 2 2
Š1 1š J 2 ª sin 2 i º 3 2
(2)
Where Re is Earthâ&#x20AC;&#x2122;s equatorial radius and J2 = 0.00108263 is a constant related to Earthâ&#x20AC;&#x2122;s oblateness. Substituting Eq. 2 in Eq. 1 and noting that y R J2 y R J2 3 sin i cos i y R J2 3e " R J2 , " RJ R J 2, " 2 1 1 2 ya a y i ye 1 e sin 2 i 2 s y R J2 y R J2 y R J2 and " " " 0 , we obtain the following yM y\ y< analytical relationships for orbital elements of the satellite: ÂŻ ² ² ² ² ² ² ° ² ² ² ² ² ² Âą
a " 0 q a(t ) " a0 " cte e " 0 q e(t ) " e0 " cte i " 0 q i(t ) " i0 " cte 2  Ÿ Š Re š 3 <(t ) " <0  cos i ½ t t0 nJ º 2ª 2 2  2(1 e ) ½  a Ž ž 2 Ÿ  ŠR š 3 \ (t ) " \ 0  nJ 2 ª e º 5 cos 2 i 1 ½ t t0 2 2  4(1 e ) ½  a ž Ž M ( t ) " M 0 n ( t t0 )
Where
2 ÂŹ ŠR š 3 n " M " n0 Â1 J 2 ÂŞ e Âş 3cos 2 i 1 3  a ÂŤ Âť ÂÂŽ 4(1 e2 ) 2
(3)
Âź
½½ .
With the
½ž
orbital elements determined at any time t, satellite I position, i.e. rS , may be determined in the geocentric inertial frame as: ď ˛ rS (t ) =
Â&#x2022; cos 1 cos t < sin 1 sin t cos i < cos 1 sin t < sin 1 cos t c a(1 < e2 ) Âł sin t + cos 1 cos t c Âł sin 1 cos t + cos 1 sin t cos i < sin 1(4) 1 + e cos Â&#x17E; (t ) Âł t sin i sin sin i cos t Â&#x2013;
< cos 1 sin t < sin 1 cos t cos i sin 1 sin i < sin 1 sin t + cos 1 cos t cos i < cos 1 sin i sin i cos t cos i
Â&#x2014; Â&#x2022; cos Â&#x17E; (t ) ¾³ Âľ Âł sin Â&#x17E; (t ) ÂľÂ&#x2DC; ÂłÂ&#x2013; 0
Â&#x2014; Âľ Âľ Âľ Â&#x2DC;
And Ď&#x2018;(t), i.e. satellite true anomaly, is determined from Keplerâ&#x20AC;&#x2122;s equation by iterative methods such as NewtonRaphson. In case of circular orbits, simply Ď&#x2018;(t)=M(t). With the satellite position determined at any time t, now ground segment position vector in the inertial frame must be determined. Based on WGS84 model, Fig. 2 shows a
J. Aerosp.Technol. Manag., SĂŁo JosĂŠ dos Campos, Vol.3, No.1, pp. 53-58, Jan. - Apr., 2011