Cross-Track Variations:AcknowledgementRadial, In-Track, and Cross-Track Variations Due to Perturbations inthe Initial Conditions of Near Circular OrbitsASEN 32004/15/02George H. BornIntroductionIt is often of interest to determine how two orbits differ because of small perturbations in the initial conditions. We will develop the expressions for the first order radial, in-track, and cross-track (RIC) variations with time between two near-circular, two-body orbits, given differences in their initial values of a, e, i, , , and .Variations in the Radial Direction, RVariations in r, the satellite geocentric radius, can be determined from 1 cosr a e EA , (1)where EA is the eccentric anomaly. Differentiating Eq. (1) using the chain rule yieldsr r rdr da de dEAa e EA      . (2)Let the differential assume a small but finite magnitude so that, for example, dr r, then  1 cos cos sinr a e EA a e EA ae EA EA      . (3)We will assume a near circular orbit and generally ignore terms of O(e) and( )i jO  , where  is a variation in any of the orbit elements. We have retained a few of the larger terms proportional to e. This is because a perturbation in a changes the mean motion which introduces a secular change in true anomaly, .It can be shown that (Roy, 1965)( ).EA O e (4)Hence, in Eq. (3) we can use,EA;and (5)EA ;.1The expression for  is derived in the section on variations in the in-track direction andis given by Eq. (17) i.e.,0 0( )n t t     (6)where we have dropped the term, 2ecosM, and n is given by Eq(16). The first term in Eq. (6) represents a perturbation in the initial value of  and the second term is due to a perturbation in a which will cause a change in mean motion and hence a change in .Equation (3) becomes 1 cos cos sin .r a e a e ae         Hence, a small change in a yields 01 cos ( )sinr a e ae n t t      . (7)A variation in e yieldscos sinr a e ae      , (8)where the last term is included because a variation in e causes a variation in  which is given by (see appendix A)2 sine   . (9)Substituting Eq. (9) into Eq. (8) yields,2cos 2 sinr a e ae e     . (10)In summary, Eqs. (7) and (10) plus the contribution of 0 yield the dominant variation in the radial direction, i.e.,  20 01 cos ( ) sin cos 2 sin sinR a e ae n t t a e ae e ae               . (11)There is a second order contribution to R caused by a variation in true anomaly as shown in the sketch below.Here r* is the reference radius and rp is the perturbed radius. The projection of the perturbed radius on r* is rcos. Hence, there is a contribution to R given by2r*rp 1 cosR r  .Because a perturbation in a causes a secular change in , this term becomes more important with time. We will ignore the contribution of 0 and  due to e here since they are small. We can write the equation for R by using Eq. (6) for , i.e.,   01 cosR r n t t   , (12)where n is given by Eq (16) and the reference orbit radius is used in Eq. (12).Note that perturbations in i, , and  to first order do not contribute to R.Variations in the In-Track Direction, IPerturbations in the argument of latitude,,u  will cause in-track variations in the orbit. Variations in u will be caused by direct perturbations in  or  at the epoch time or by perturbations in a and/or e which will cause subsequent variations in the argument of latitude since they influence .The value of I, the in-track variation,is given by 1 cosI r ua e M u   (13)where,u    .If we ignore terms of O(e) we can write 0 0M . (14)Also2 sinM e M .So 1 2 cose M M   . (15)Using3orbiturequator 0 0M M n t t  yields 0 0M M n t t    where1/ 23/ 2naand3/ 2 .nn aa  (16)Hence, using Eqs. (14), (15) and (16)    0 01 2 cose M n t t       (17)Also, a perturbation in e affects  (see Appendix A) and2 sin .e   (18)Substituting Eqs. (14), (17), and (18) into Eq. (13) and including the contribution of  yields,     0 01 2 cos 2 sinI r e M n t t e           . (19)Note that      1 2 cos 1 cos 1 2 cos1 cosr e M a e M e Ma e M   ;Hence,      0 02 sin 1 cosI r e a e M n t t           (20)In addition, a perturbation in the right ascension of the ascending node, , causes both anin-track and a cross-track variation as seen in the sketch below.4IiCReferenceorbitPerturbedorbitiICFrom spherical trig:sin sin sin .C i Using the small angle approximation,sin .C iThis will be the cross-track error at the equator. However, C will vary with argument oflatitude and a positive value for  will produce a negative value of C at the equator. Hence,cosC r C usin cosC r i u (21)Also, for the right spherical trianglecos tan cot ,i I orcostan .cotiI(22)Using the small angle approximationcos .I i(23)Hence, the in-track contribution due to a variation in  iscos .I r Ir i (24)5Cross-Track Errors, CWe have already seen that a perturbation in  yields a cross-track error given by Eq. (21). The other orbit element which contributes to a cross-track error is the inclination. This is illustrated in the sketch below:SummaryAs stated earlier, we have ignored all second order effects in i j  . For example if we perturb  this will affect I and C. If we then perturb i there will be a first order change in C. In addition, there will be second order changes in both i and C. These changes willbe proportional to various products of , , , andi     and are ignored here. The results derived here can be summarized as follows (Note that we have substituted M for): Radial Variations: 0203(1