Visibility intervals between two artificial satellites under the action of Earth oblateness

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Abstract

This paper presents an analytical method to determine the rise-set times of satellite-satellite visibility periods in different orbits. The Visibility function in terms of the orbital elements of the two satellites versus the time were derived explicitly up to e4. The line-of-sight corrected for Earth Oblateness up to J2, were considered as a perturbation to the orbital elements. The visibility intervals of the satellites were calculated for some numerical examples in order to test the results of the analytical work.

Abstract

This paper presents an analytical method to determine the rise-set times of satellite-satellite visibility periods in different orbits. The Visibility function in terms of the orbital elements of the two satellites versus the time were derived explicitly up to e4. The line-of-sight corrected for Earth Oblateness up to J2, were considered as a perturbation to the orbital elements. The visibility intervals of the satellites were calculated for some numerical examples in order to test the results of the analytical work.

1 Introduction

The long term and highly accurate orbit estimation, especially rise-and-set time computation, plays a key part in the pre-request information for mission analysis and on-board resources management in more general communication, Earth observation and scientific spacecraft, there has been a big trend to use low cost, fast access and multi-functional small satellites to provide and exchange information for a wide range of military and civil applications such as communication in a very remote area for changing conditions, low cost store-and-forward communication, disaster warning for global shipping service and some Earth Observation missions. Therefore, this requires accurate estimation of when the satellites will start to be visible (rise) to a given location on the Earth or to other satellite and the time when the satellite disappears from the horizon (set) over a time-scale of months in some cases.

The rise/set problem may be defined as the process of determining the times at which a satellite rises and sets with respect to a ground location. The easiest solution uses a numerical method to determine visibility periods for the site and satellite by evaluating UK position vectors of each. It advances vectors by a small time increment, Δt, and checks visibility at each step. A drawback to this method is computation time, especially when modeling many perturbations and processing several satellites. Escobal [1], [2] proposed a faster method to solve the rise/set problem by developing a closed-form solution for unrestricted visibility periods about an oblate Earth. He assumes infinite range, azimuth, and elevation visibility for the site. Escobal transforms the geometry for the satellite and tracking station into a single transcendental equation for time as a function of eccentric anomaly. He then uses numerical methods to find the rise and set anomalies, if they exist. Lawton [3] has developed another method to solve for satellite-satellite and satellite-ground station visibility periods for vehicles in circular or near circular orbits by approximating the visibility function, by a Fourier series.

More recently, Alfano, Negron, and Moore [4] derived an analytical method to obtain rise/set times of a satellite for a ground station and includes restrictions for range, azimuth, and elevation. The algorithm uses pairs of fourth-order polynomials to construct functions that represent the restricted parameters (range, azimuth, and elevation) versus time for an oblate Earth. It can produce these functions from either uniform or arbitrarily spaced data points. The viewing times are obtained by extracting the real roots of localized quantic.

Palmar [5], introduced a new method to predict the passes of satellite to a specific target on the ground which is useful for solving the satellite visibility problem. he firstly described a coarse search phase of this method including two-body motion, secular perturbation and atmospheric drag, then he described the second phase (refinement), which uses a further developed controlling equation F (a) = 0 based on the epicycle equations

In this work, a fast method for satellite-satellite visibility periods for the rise-and-set time prediction for two satellites in terms of classical orbital elements of the two satellites versus time were established. The secular variations of the orbital elements due to Earth Oblateness were taken into account in order to consider the changes in the nodal period of satellite and the changes in the long term prediction of maximum elevation angle. In the following description we will introduce the formulae for satellite rise-and-set times of the two satellites. The derived visibility function provides high accuracy over a long period, and provides direct computation of rise-set times.

2 Visibility Analysis

The location of a satellite is determined by the Kepler’s laws, in addition, orbit perturbations due to Earth Oblateness are considered. A set of six orbital parameters is used to fully describe the position of a satellite in a point in space at any given time: semi-major axis a, eccentricity e, inclinations of the orbit plane i, right ascension of the nodeΩ, the argument of perigee ω, and true anomaly f. The above parameters are shown in the Figure1.

Fig. 1

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Fig. 1

Geometry of Satellites Visibility

Citation: Applied Mathematics and Nonlinear Sciences 3, 2; 10.21042/AMNS.2018.2.00028

The links between two satellites are determined by the visibility analysis presented as follows:

Referring to Fig.1, the position vectors of satellites 1, and 2 with respect to the ECI coordinate system are r1and r2.The position vector from satellite 1 to satellite 2 will be denoted by ρ=r2r1.

Let h = OP = Reh, be the perpendicular from the dynamical center of the earth to the range vector , cutting the earth surface at Q and the range vector ρ,where Re is the mean radius of the Earth, and Δh is the thickness of the atmosphere above the surface of the Earth to P. From the geometry of Fig. 1, we have the following relations:

AreaofΔOS1S2=12|r1r2|=12|r1||r2|sinψ

On the other hand, the area can be calculated from the relation:

AreaofΔOS1S2=12|r2r1|h

From Eqs. 1 and 2, we conclude that:

h=Re+Δh=|r1||r2|sinψ|r2r1|

Where ψ is the angle between r1andr2the two satellites. The condition for direct visibility between the two satellites can be determined from Eq. 3 by putting

Δh=|r1||r2|sinψ|r2r1|Re

Where

ψ=cos1(r1r2|r1||r2|)

After the determination of the locations of satellites at any time in the space, they can only achieve visibility when they are both above the same plane which is tangent to the earth surface. The extreme situation is that both of them are in the tangent plane.

When Δh > 0, the two satellites can achieve visibility. Otherwise, there is no visibility.

Δh>0:Δh=|r1||r2|sinψ|r1r2|Re=r12r22(r1r2)2(r12+r22)2(r1r2)Re

i.e. When

r12r22(r1r2)2(r12+r22)2(r1r2)Re2

After rearrangement the terms, we obtain:

r12r22(r1r2)2Re2[(r12+r22)2(r1r22)2(rr2)]

Therefore, the visibility function, V, that describes whether these two satellites can achieve visibility is gained, as follows:

V=Re2[(r12+r22)2(r1r2)]r12r22+(r1r2)2

Where

V={+ve,Nonvisibilitycase0,riseorsetve,directlineofsight

If the position relation between two satellites satisfies the visibility conditions, two satellites can communicate with each other over interstellar links.

3 Construction of The Visibility Function

The position vector of each satellite in the geocentric coordinate system,r=(x,y,z), can be calculated by the following formula [6]:

(xyz)=r(cosΩcos(ω+f)sinΩsin(ω+f)cosisinΩcos(ω+f)+cosΩsin(ω+f)cosisin(ω+f)sini)

Where r denote the distance from the earth center O to the satellite, given by:

r=a(1e2)1+ecosf

Forming scalar product (r1r2),keeping terms up to O(e4) only, we obtain

r1r2=x1x2+y1y2+z1z2

For the sake of simplification of calculations, we put the coordinates of the satellite as:

x1=r1[σ12cos(f1+ω1+Ω1)+γ12cos(f1+ω1+Ω1)]y1=r1[σ12sin(f1+ω1+Ω1)+γ12sin(f1+ω1+Ω1)]z1=2r1σ1γ1cos(f1+ω1)

Where σ1 = cos (i1/2) and γ1 = sin (i1/2), with similar expressions for the other satellite. In order to obtain the visibility function as an explicit function of time, we transform the true anomaly f, to the mean anomaly M, using the following transformation formulas Brouwer [6] up to O(e4)

r1=a1[(1+12e12)+(e1+38e13)cosM1+(12e12+13e14)cos2M138e13cos3M113e14cos4M1]
r1cosf1=a1[32e+(138e12+5192e14)cosM1(12e1+12e13)cos2M1+(12e12+13e14)cos2M1+(38e1245128e13)cosM1+13e13cos4M1]
r1sinf1=a1158e1211192e14sinM1+12e1512e13sin2M1+38e1251128e14sin3M1+13e13sin4M1

With similar expressions for the other satellite.

Substituting Eqs. ( 6–8 ) into Eq. 5, and keeping terms up to O(e4), we obtain:

x=a24σ23e2+e4cos(MωΩ)+e3cos(2MωΩ)+916e4cos(3MωΩ)+122e2cos(M+ω+Ω)+12e9e3cos(2M+ω+Ω)+9e2e4cos(3M+ω+Ω)+8e4cos(4M+ω+Ω)36ecosω+Ω+γ22412e238e4cos(M+ωΩ)+12e9e3cos(2M+ωΩ)+9e2e4cos(3M+ωΩ)+8e4cos(4M+ωΩ)+3e2+e4cos(Mω+Ω)+e3cos(2Mω+Ω)916e4cos(3Mω+Ω)36ecosωΩ
y=a24σ23e2e4sin(MωΩ)e3sin(2MωΩ)916e4sin(3MωΩ)+2412e238e4sin(M+ω+Ω)+12e9e3sin(2M+ω+Ω)+9e2e4sin(3M+ω+Ω)+8e4sin(4M+ω+Ω)36esinω+Ω+γ224+12e2+38e4sin(M+ωΩ)+12e+9e3sin(2M+ωΩ)+9e2+e4sin(3M+ωΩ)8e4sin(4M+ωΩ)+3e2+e4sin(Mω+Ω)+e3sin(2Mω+Ω)+916e4sin(3Mω+Ω)36esinωΩ

Using Eq. 6, we can expand the term r12r22in Eq. 4 up to the 4th degree in the eccentricities as:

r12r22=a12a2244+4e12+4e22+4e12e22+e14+e248e1+4e13+12e1e22cos(M1)8e2+4e23+12e12e2cos(M2)+2e12e12e22cos(2M1)+2e22e12e22cos(2M2)e13e2cos3M1M212e12e22cos(2M12M2)12e12e22cos(2M1+2M2)+8e1e2+3e13e2+3e1e23cos(M1+M2)e1e23cosM13M2e13e2cos3M1+M2+8e1e2+3e13e2+3e1e23cos(M1M2)e1e23cosM1+3M2

Scalar product in Eq. 4 can be formed by using Eq. 8, we obtain:

r1r2=a1a232i=011Ti
T0=19e13e2+19e1e23
T1=72e1e2+8e1e2s12+8e1e2s22cosω1ω2+48e28e2e12+2e23+16s12+16s22cos(M1+ω1ω2)24e1e2+27e13e2+13e1e23+8e1e2s12+8e1e2s22cos(2M1+ω1ω2)+10e12e2cos3M1+ω1ω2+13e13e2cos(4M1+ω1ω2)18e23cos(M14M2+ω1ω2)+22e1e22cos2M13M2+ω1ω2
T2=+13e1e23cos2M14M2+ω1ω2+18e12e22cos3M13M2+ω1ω2+20e22+22e12e22+18e24cosM13M2+ω1ω2+2e22e12e22cos(2M2)+18e1e22cos(M2+ω1ω2)+16e2+14e238e12e22+16e2s12+16e2s22cos(M12M2+ω1ω2)+8e1e2+5e13e2+5e1e238e1e2s128e1e2s22cos(2M12M2+ω1ω2)
T3=e13e2cos3M1+M2+ω1ω222e2e12cos3M12M2+ω1ω2+13e2e13cos4M12M2+ω1ω218e13cos(4M1M2+ω1ω2)+12e222e24+2e12e22cos(M1+M2+ω1ω2)+3216e1216e22+16e12e2232s1232s22cosM1M2+ω1ω2+16e1+14e13+24e1e22+16e1s12+16e1s22cos(2M1M2+ω1ω2)+20e12+18e14+6e12e22cos(3M1M2+ω1ω2)
T4=16e1+18e13+24e1e2216e1s1216e1s22cos(M2ω1+ω2)+12e22+2e12e222e24cos(M1+M2+ω1ω2)+14e1e22cos(2M1+M2+ω1ω2)
T5=18e1e22cos(M2+ω1ω2)5e1e23cos(2M2+ω1ω2)18e12e2cos(M1ω1+ω2)+2e12e22cos(M1M2ω1+ω2)+12e12+18e12e222e14cos(M1+M2ω1+ω2)
T6=+40e1e2+2e13e2+5e1e238e1e2s128e1e2s22cos(2M2ω1+ω2)5e13e2cos(2M1+2M2ω1+ω2)+2e13cos(2M1+M2ω1+ω2)18e12e2cos(M1+2M2ω1+ω2)16e1s12cos(M2+ω1+ω22Ω1)
T7=+8e1e2s12cos(ω1+ω22Ω1)+13e1e23cos(4M2ω1+ω2)+10e12e22cos(M1+3M2ω1+ω2)16e1s12cos(2M1+M2+ω1+ω22Ω1)16e2s12cos(M1+ω1+ω22Ω1)+8e1e2s12cos(2M1+ω1+ω22Ω1)+(32s12+16e22s12+16e12s12)cos(M1+M2+ω1+ω22Ω1)22e1e22cos(3M2ω1+ω2)
T8=16e1s22cos(2M1+M2+ω1+ω22Ω2)+8e1e2s22cos(2M2+ω1+ω22Ω2)16e2s22cos(M1+2M2+ω1+ω22Ω2)+8e1e2s22cos(2M1+2M2+ω1+ω22Ω2)16e1e2s1s2cos(ω1+ω2Ω1Ω2)+32e2s1s2cos(M1+ω1+ω2Ω1Ω2)16e1e2s1s2cos(2M1+ω1+ω2Ω1Ω2)+32e1s1s2cos(M2+ω1+ω2Ω1Ω2)
T9=+8e1e2s12cos(2M1+ω1+ω22Ω1)16e2s12cos(M1+2M2+ω1+ω22Ω1)+8e1e2s12cos(2M1+2M2+ω1+ω22Ω1)+8e1e2s22cos(ω1+ω22Ω2)16e2s22cos(M1+ω1+ω22Ω2)+8e1e2s22cos(2M1+ω1+ω22Ω2)16e1s22cos(M2+ω1+ω22Ω2)+(32s22+16e12s22+16e22s22)cos(M1+M2+ω1+ω22Ω2)
T10=64s1s2+32e12s1s2+32e22s1s2cos(M1+M2+ω1+ω2Ω1Ω2)+32e1s1s2cos(2M1+M2+ω1+ω2Ω1Ω2)16e1e2s1s2cos(2M2+ω1+ω2Ω1Ω2)+32e2s1s2cos(M1+2M2+ω1+ω2Ω1Ω2)16e1e2s1s2cos(2M1+2M2+ω1+ω2Ω1Ω2)
T11=32e1s1s2cos(M2ω1+ω2+Ω1Ω2)+16e1e2s1s2cos(ω1ω2Ω1+Ω2)32e2s1s2cos(M1+ω1ω2Ω1+Ω2)+16e1e2s1s2cos(2M1+ω1ω2Ω1+Ω2)32e2s1s2cos(M12M2+ω1ω2Ω1+Ω2)+16e1e2s1s2cos(2M12M2+ω1ω2Ω1+Ω2)32e1s1s2cos(2M1M2+ω1ω2Ω1+Ω2)+64s1s2+32e12s1s2+32e22s1s2cos(M1M2+ω1ω2Ω1+Ω2)

Substituting Eqs. 9 and 10 into Eq. 4, we obtain the complete expression of the visibility function in terms of the orbital elements in the form:

V=i=015Vi
V1=A0+A1cos(M1)+A2cos(2M1)+A3cos(3M1)+A4cos(4M1)+A5cos(M2)+A6cos(2M2)+A7cos(3M2)+A8cos(4M2)+A9cos(M1M2)+A10cos(M12M2)+A11cos(M13M2)+A12cos(2M1M2)+A13cos(3M1M2)+A14cos(2M12M2)+A15cos(M1+M2)+A16cos(2M1+M2)+A17cos(3M1+M2)+A18cos(M1+M2)+A19cos(2M1+M2)+A20cos(M1+3M2)+A21cos(2ω12Ω1)+A22cos(2ω22Ω2)
V2=A23cos(M1+2ω12Ω1)+A24cos(2M1+2ω12Ω1)+A25cos(4M1+2ω12Ω1)+A26cos(2M12M2+2ω12Ω1)+A27cos(M1M2+2ω12Ω1)+A28cos(M1+M2+2ω12Ω1)+A29cos(2M1+M2+2ω12Ω1)+A30cos2ω12ω2+A31cos(M1+2ω12ω2)+A32cos(2M1+2ω12ω2)+A33cos(3M1+2ω12ω2)+A34cos(4M1+2ω12ω2)
V3=A35cos(2M16M+2ω12ω2)+A36cos(M15M2+2ω12ω2)+A37cos(3M15M2+2ω12ω2)+A38cos(M14M2+ω1ω2)+A39cos(3M14M+2ω12ω2)+A40cos(4M14M2+ω1ω2)+A41cos(2M14M2+2ω12ω2)+A42cos(M13M2+2ω12ω2)+A43cos(2M13M+2ω12ω2)+A44cos(3M13M2+2ω12ω2)+A45cos(4M13M2+2ω12ω)+A46cos(5M13M2+2ω12ω2)+A47cos(2M12M2+2ω12ω2)+A48cos(3M12M2+2ω12ω)
V4=A49cos(5M13M2+2ω12ω2)+A50cos(4M12M2+2ω12ω2)+A51cos(6M12M2+2ω12ω2)++A52cos(M1M2+2ω12ω2)
V5=A53cos(2M1M2+2ω12ω2)+A54cos(3M1M2+2ω12ω)+A55cos(4M1M2+2ω12ω2)+A56cos(5M1M2+2ω12ω2)+A57cos(M1+M2+2ω12ω)+A58cos(2M1+M2+2ω12ω2)
V6=A59cos(3M1+M2+2ω12ω2)+A60cos(2M1+2M2+2ω12ω)+A61cos(2ω12Ω2)+A62cos(2M12M2+2ω12ω2)+A63cos(M1M2+ω1ω2)+A64cos(M1+M2+ω1+ω2)+A65cos(M22ω1+2ω2)+A66cos(M1+M22ω1+2ω2)+A67cos(M1+3M22ω1+2ω2)+A68cos(M1+2M22ω1+2ω2)+A69cos(2M1+2M22ω1+2ω2)+A70cos(3M22ω1+2ω2)+A71cos(4M22ω1+2ω2)+A72cos(M22Ω1+2ω2)+A73cos(M1+M22Ω1+2ω2)+A74cos(2M22ω1+2ω2)
V7=A75cos(2M22Ω1+2ω2)+A76cos(M1+2M22Ω1+2ω2)+A77cos(2M1+2M22Ω1+2ω2)+A78cos(M1+3M22Ω1+2ω2)+A79cos(4M22Ω1+2ω2)+A80cos(2M1+2M2+2ω1+2ω22Ω2)+A81cos(2M1+2M2+2ω1+2ω2Ω13Ω2)+A82cos(2M2+Ω1+2ω23Ω2)+A83cos(M1+2ω12Ω2)+A84cos(2M1+2ω12Ω2)+A85cos(4M1+2ω12Ω2)+A86cos(2M12M2+2ω12Ω2)+A87cos(M1M2+2ω12Ω2)+A88cos(M1+M2+2ω12Ω2)+A89cos(2M1+M2+2ω12Ω2)+A90cos(M2+2ω22Ω2)
V8=+A91cos(M1+M2+2ω22Ω2)+A92cos(2M2+2ω22Ω2)+A93cos(2M2+2ω22Ω2)+A94cos(2M1+2M2+2ω22Ω2)+A95cos(M1+3M2+2ω22Ω2)+A96cos(4M2+2ω22Ω2)
V9=A97cos(2M1+2M2+2ω1+2ω22Ω12Ω2)+A98cos(2Ω12Ω22ω2)+A99cos(M1+2ω1Ω1Ω2)+A100cos(2M1+2ω1Ω1Ω2)+A101cos(4M1+2ω1Ω1Ω2)+A102cos(2M12M2+2ω1Ω1Ω2)+A103cos(M1M2+2ω1Ω1Ω2)+A104cos(M1+M2+2ω1Ω1Ω2)+A105cos(2M1+M2+2ω1Ω1Ω2)+A106cos(2M1+2M2+2ω1Ω1Ω2)+A107cos(Ω1Ω2)+A108cos(2M1+Ω1Ω2)+A109cos(M2+Ω1Ω2)
V10=A110cos(2M2+Ω1Ω2)+A111cos(ω1ω2+Ω1Ω2)+A112cos(M1+ω1ω2+Ω1Ω2)+A113cos(2M1+ω1ω2+Ω1Ω2)+A114cos(3M1+ω1ω2+Ω1Ω2)+A115cos(M14M2+ω1ω2+Ω1Ω2)+A116cos(2M13M2+ω1ω2+Ω1Ω2)+A117cos(3M13M2+ω1ω2+Ω1Ω2)+A118cos(M12M2+ω1ω2+Ω1Ω2)+A119cos(M12M2+ω1ω2+Ω1Ω2)+A120cos(3M12M2+ω1ω2+Ω1Ω2)+A121cos(2M12M2+ω1ω2+Ω1Ω2)+A122cos(M1M2+ω1ω2+Ω1Ω2)+A123cos(2M1M2+ω1ω2+Ω1Ω2)+A124cos(3M1M2+ω1ω2+Ω1Ω2)
V11=A125cos(4M1M2+ω1ω2+Ω1Ω2)+A126cos(M1+M2+ω1ω2+Ω1Ω2)+A127cos(2M1+M2+ω1ω2+Ω1Ω2)+A128cos(3M1+M2+ω1ω2+Ω1Ω2)+A129cos(2M2+ω1ω2+Ω1Ω2)+A130cos(M1+2M2+ω1ω2+Ω1Ω2)
V12=A131cos(2M1+2M2+ω1ω2+Ω1Ω2)+A132cos(M1+3M2+ω1ω2+Ω1Ω2)+A133cos(ω1+ω2+Ω1Ω2)+A134cos(M1+ω1+ω2+Ω1Ω2)+A135cos(2M1+ω1+ω2+Ω1Ω2)+A136cos(M1M2+ω1+ω2+Ω1Ω2)+A137cos(2M1+M2+ω1+ω2+Ω1Ω2)+A138cos(M2+ω1+ω2+Ω1Ω2)+A139cos(3M1+M2+ω1+ω2+Ω1Ω2)+A140cos(2M2+ω1+ω2+Ω1Ω2)
V13=+A141cos(M1+2M2+ω1+ω2+Ω1Ω2)+A142cos(2M1+2M2+ω1+ω2+Ω1Ω2)+A143cos(M1+3M2+ω1+ω2+Ω1Ω2)+A144cos(2M1+2M2+ω1+ω2+Ω1Ω2)+A145cos(M1M2+ω1ω2Ω1Ω2)+A146cos(M1M2ω1ω2+Ω1+Ω2)+A147cos(M2ω1+ω2Ω1+Ω2)+A148cos(M1+M2ω1+ω2Ω1+Ω2)+A149cos(2M1+M2ω1+ω2Ω1+Ω2)+A150cos(3M1+M2ω1+ω2Ω1+Ω2)
V14=A151cos(2M2ω1+ω2Ω1+Ω2)+A152cos(M1+2M2ω1+ω2Ω1+Ω2)+A153cos(2M1+2M2ω1+ω2Ω1+Ω2)+A154cos(3M2ω1+ω2Ω1+Ω2)+A155cos(M1+3M2ω1+ω2Ω1+Ω2)+A156cos(ω1+ω2Ω1+Ω2)+A157cos(M1+ω1+ω2Ω1+Ω2)+A158cos(2M1+ω1+ω2Ω1+Ω2)
V15=+A159cos(M2+ω1+ω2Ω1+Ω2)+A160cos(M1+3M2+ω1+ω2Ω1+Ω2)+A161cos(2M1+M2+ω1+ω2Ω1+Ω2)+A162cos(M1+2M2+ω1+ω2Ω1+Ω2)+A163cos(2M1+2M2+ω1+ω2Ω1+Ω2)+)A164cos(M1+M2+ω1+ω2Ω1+Ω2.

Where the coefficients As, s = 0,1, …,164 are functions of (aj , ej , i j), j = 1,2 are given in Appendix A.

4 Adding Perturbing Forces

We shall consider the effect of perturbation on the orbital elements due to the Earth Oblateness. So, the orbital elements of the two satellites can be written in the form:

σj(t)=σjo+Δσjoblj=1,2.

Where(Δσj)obl denote the first order perturbation in the orbital elements, and j = 1, 2 denotes satellites 1 and 2.

The expansion of the perturbed visibility function about some epoch time t0 can be obtained by Taylor expansions about the osculating elements (σ0j,e0j,i0j,Ω0j0j,M0j) up to the first order as:

F(aj,ej,ij,Ωj,ωj,Mj)=V(a0j,e0j,i0j,Ω0j,ω0j,M0j)+s=16Vσs0Δσs

The symbols σs, represent any of the orbital elements. The summation ranges from s = 1 to 3 represents the elements (Ω11,M1) and from s = 4 to 6 represents (Ω22,M2) respectively. The quantities Δσs, s = 1, 2,..,6 represent the secular variations in the corresponding orbital elements due to the perturbation.

4.1 The effect of Earth Oblateness

A satellite under the influence of an inverse square gravitational law has truly constant orbital elements. In reality, however, there is a gradual change in the orbital elements due to the Earth’s Oblateness. The principal effect of this is to introduce a short period oscillation of the orbital elements, which we can ignore in most cases. The argument of perigee ω, the longitude of the ascending node Ω , and the Mean anomaly M, however, experience a secular drift which significantly changes the long term prediction of maximum elevation angle. Using the method of variation of parameters to take proper account of all these secular variations due to earth oblateness up to J2. The perturbation method is explained in many standard textbooks on [7].

The gravitational potential, U, of a satellite including the contribution of J2, is given by [7]

U=μr1+J2Re22r2(13sin2δ)

Which may be expressed as:

U=U0+R

Where

U: the potential of the Earth,

U0: the potential of purely spherical Earth,

R : the perturbing function,

μ = k2m: the gravitational constant× mass of the Earth,

J2:Coefficient of the 2th harmonic,

δ: the satellite altitude.

Since the equation of the satellite orbit is

r=a(1e2)1+ecos(f)

From spherical trigonometry, we have the relation

sin(δ)=sin(i)sin(f+ω)

Then the perturbing potential to the order of J2 will take the form:

R=32μJ2Re2r31312sin2i+12sin2icos2(f+ω)

Where

a : is the semi-major axis of the orbit,

e : is the orbital eccentricity,

f : is the true anomaly,

i : is the orbit inclination,

ω: is the argument of perigee.

Since we consider only the secular variation, so we average the perturbative function, with respect to the mean anomaly M . The derivation and solution are given in many text books for example [8].

a(t)=a0,e(t)=e0,i(t)=i0Ω¯(t)=Ω0+Ω˙¯(tt0),ω¯(t)=ω0+ω˙¯(tt0),M¯(t)=M0+n¯(tt0)+Δn¯(tt0)

Finally, secular variations are associated with a steady non-oscillatory, continuous drift of an element from the adopted epoch value,

ΔΩ¯=Ka2η412γ2n¯tt0Δω¯=2Ka2η415σ2γ2n¯tt0ΔM¯=1+Ka2η316σ2γ2n¯tt0

Where,

n¯=μ/a3,K=32J2Re2,η=1e2

Adding Perturbations

According the Eq. 11, we can add the effects of two perturbations due to Oblateness on the six elements of orbits together as:

ω(t)=ω0+(Δω)oblΩ(t)=Ω0+(ΔΩ)oblM(t)=M0+(ΔM)obl

5 Numerical Examples

In what follows the visibility function were tested for some examples to obtain the mutual visibility between two Earth Satellites whatever the types of their orbits may be. Classical orbital elements for some satellites from the Center for Space Standards & Innovation were used as test data for this study, and are listed in Tables 1 and 2.

Table 1

Norad Two - Line Element Sets For The Satellites AQUA, ARIRANG-2, HST and ODIN

Satellite Orbital Elements1-AQUA2-ARIRANG-23-HST4-Odin
Equivalent altitude (Km)699.588682.6205543.2687540.5256
a (Km)7077.7257060.7576921.4056918.662
n (rev/min)0.0104080.0104450.0107790.010769
e0.0002860.0016690.0002560.001057
i (degree)98.203198.067628.470597.591
Ω (degree)121.609776.990617.611200.4958
ω (degree)54.081258.4665301.12186.4076
M (degree)125.1605101.4671170.9719173.7019
ρ (kg/km3)3.63E-054.6E-050.0003540.000369
ρo (kg/km3)0.0001450.0001450.0006970.000697
h0(Km) (kg/km3)600600500500
H (Km)71.83571.83563.82263.822
Epoch Year & Julian Date18180. 5977074918180. 8201966518182.93559318182.93790454
time of data (min)2018 06 292018 06 292018 07 012018 07 01
13:31:3019:41:03.00421:57:32.13422:30:32.994

The visibility intervals are shown in the following Figures 2,4,6 without any perturbing force and with Oblateness force in Figures 3, 5, 7, calculated in Tables 1, 2 and 3, respectively.

Fig. 2

Download Figure

Fig. 2

Visibility Intervals Between AQUA and ARIRANG2 during 24-H

Citation: Applied Mathematics and Nonlinear Sciences 3, 2; 10.21042/AMNS.2018.2.00028

Fig. 3

Download Figure

Fig. 3

Visibility Intervals Between AQUA and ARIRANG2 24-H with Oblateness Force

Citation: Applied Mathematics and Nonlinear Sciences 3, 2; 10.21042/AMNS.2018.2.00028

Fig. 4

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Fig. 4

Visibility Intervals Between HST and ODIN For 24-H

Citation: Applied Mathematics and Nonlinear Sciences 3, 2; 10.21042/AMNS.2018.2.00028

Fig. 5

Download Figure

Fig. 5

Visibility Intervals Between HST and ODIN For 24-H with Oblateness Force

Citation: Applied Mathematics and Nonlinear Sciences 3, 2; 10.21042/AMNS.2018.2.00028

Fig. 6

Download Figure

Fig. 6

Visibility Intervals Between CFESAT and MTI For 24-H

Citation: Applied Mathematics and Nonlinear Sciences 3, 2; 10.21042/AMNS.2018.2.00028

Fig. 7

Download Figure

Fig. 7

Visibility Intervals Between CFESAT and MTI For 24-H with Oblateness Force

Citation: Applied Mathematics and Nonlinear Sciences 3, 2; 10.21042/AMNS.2018.2.00028

Table 2

Norad Two - Line Element Sets For The Satellites CFESAT and MTI

Satellite Orbital Elements5-CFESAT6-MTI
Equivalent altitude (Km)468.8831412.5092
a (Km)6847.026790.646
n (rev/min)0.0109530.011074
e0.0005820.000812
i (degree)35.424797.5789
Ω (degree)203.04317.7612
ω (degree)183.8662345.6071
M (degree)176.2019143.5229
ρ (kg/km3)0.0011620.003008
ρo (kg/km3)0.0015850.003725
h0(Km) (kg/km3)450400
H (Km)60.82858.515
Epoch Year & Julian Date18182.501732218182.7746284
time of data (min)2018 07 012018 07 01
12:02:28.52618:02:08.608
Table 3

Visibility Intervals Between AQUA and ARIRANG 24 Houres

Without Earth Oblateness ForceWith Earth Oblateness Force
RiseSetvisibility timeRiseSetvisibility time
msms
117.107444.86632745.534-2.35384--
266.408694.04822738.37614.453451.67193713.1
3115.669143.3182738.9464.843100.955366.72
4164.973192.4992731.56114.976150.2223514.76
5214.236241.7682731.92164.951199.5073433.36
6263.542290.9482724.36214.755248.7543359.94
7312.807340.2162724.54264.522298.0423331.2
8362.115389.3962716.86314.144347.279338.1
9411.382438.6632716.86364.522396.57322.88
10460.692487.842278.94413.301445.83229.94
11509.962537.108278.76462.871495.0933213.32
12559.276586.286276512.306544.32320.84
13608.547635.552270.3561.825593.6153147.4
14657.863684.7292651.96611.207642.8393137.92
15707.135733.9942651.54660.689692.1353126.76
16756.453783.1712643.08710.033741.3573119.44
17805.728832.4352642.42759.488790.6553110.02
18855.048881.6112633.78808.804839.876314.32
19904.325930.8742632.94858.238889.1743056.16
20953.647980.0492624.12907.532938.3943051.72
211002.931029.312622.8965.95987.6943044
221052.251078.492614.41006.231036.913040.8
231101.351127.752613.21055.631086.213034.8
241150.861176.92263.61104.91135.433031.8
251200.141226.18262.41154.291184.733026.4
261249.471275.362553.41203.541233.953024.6
271298.751324.622552.21252.931283.253019.2
281348.081373.792542.61302.171332.473018
291397.371423.052540.81351.551381.773013.2
30----1400.791430.993012

6 Conclusions

An analytical method for the rise -set time prediction for two satellites were derived through a visibility function in terms of classical orbital elements of the two satellites versus time. The secular variations of the orbital elements due to Earth Oblateness were taken into account in order to consider the changes in the nodal period of satellite and the changes in the long term prediction of maximum elevation angle.

In the Table 3: The Visibility Intervals Between AQUA and ARIRANG 2, it is noticed from the first column of (The Function of Visibility without any perturbation) that the time of visibility periods oscillates in a periodic fashion till the ninth period then it decreases gradually. Also, from the second column we conclude that the effect of Earth Oblateness is the increasing the number of periods significantly. But the time intervals of the visibility decreases gradually.

In the Table 4: The Visibility Intervals Between HST and ODIN, It is noticed from the first column of (The Function of Visibility without any perturbation) that the time of visibility periods oscillates in a periodic manner, and then it decreases gradually. It is also noticed that the second column explains the Earth Oblateness which affects the number of period’s increases significantly and clearly. It is also observed that the time of periods of the visibility ascending increases till the ninth period then it increases and decreases in an oscillating manner and over time, stability occurs and the time period stabilizes.

Table 4

Visibility Intervals Between HST and ODIN 24 Houres

Without Earth Oblateness ForceWith Earth Oblateness Force
visibility timevisibility time
RiseSetmsRiseSetms
139.325544.443257.062-5.86362--
287.203592.0758452.33814.139722.0401754.024
3134.82139.91355.5228.846343.62881446.59
4182.699187.545450.7657.92791.35913325.926
5230.316235.38354.02105.468139.0843336.96
6278.195283.014449.14153.145186.8323341.22
7325.812330.85252.4200.854234.5593342.3
8373.691378.84359.12248.578282.3083343.8
9421.307426.32250.9296.07330.0363357.96
10469.188473.952445.84344.041377.7863344.7
11516.803521.792459.34391.776425.5143344.28
12564.684569.421444.22439.514473.2633344.9
13612.298617.261457.72487.252520.923340.08
14660.18664.98448534.992568.7413344.94
15707.794712.731456.22582.731616.473344.43
16755.676760.359440.98630.473664.2193344.76
17803.289808.201454.72678.212711.9483344.16
18851.172855.828439.18725.954759.6973344.58
19898.785903.67453.1773.694807.4263343.92
20964.668951.297437.74821.437855.1753344.28
21994.28999.14451.6869.178902.9043343.65
221042.161046.77436.6916.92950.6533343.98
231089.781094.61449.8964.661998.3823343.62
241137.661142.23434.21012.41046.133343.8
251185.271190.08448.61060.151093.863342.6
261233.161237.7432.41107.891141.613343.2
271280.771285.55446.81155.631189.343342.6
281328.651333.17431.21203.371237.093343.2
291376.261381.02445.61251.111248.823342.6
301424.151428.64429.41298.861332.563342
31----1346.61380.293341.4
32----1394.341428.043342

In the Table 5: The Visibility Intervals Between CFESAT and MTI, It is noticed from the first column of (The Function of Visibility without any perturbation) that the time of visibility periods has gradually obvious increases. Also the periods’ time of the visibility function increases significantly and clearly. The second column explains the Earth Oblateness which affects the number of periods increase significantly and clearly. It is also observed that the time of periods of the visibility increases and decreases in an oscillating manner.

Table 5

Visibility Intervals Between CFESAT and MTI 24 Houres

Without Earth Oblateness ForceWith Earth Oblateness Force
visibility timevisibility time
RiseSetmsRiseSetms
1430.519432.975227.362.4872910.1979 742.6366
2476.988479.867252.7427.835340.9145134.752
3523.051527.14745.7649.14156.6622 731.26
4569.583573.973423.474.60587.78451310.77
5615.801621.095517.6495.8712103.138 716.008
6662.353667.899532.76121.338134.6111316.38
7708.642714.947618.3142.528149.665 78.22
8755.203761.74632.22168.1181.4371320.22
9801.535808.742712.42189.261196.185 655.44
10848.101855.526725.5214.827228.2271324
11894.462902.49682.04235.913242.757 650.64
12941.03949.275814.7261.583275.0271326.64
13987.413996.22848.42282.643289.308 639.9
141033.981042.9990.6308.306321.7961329.4
151080.381089.92932.4329.285335.916 637.86
161126.951136.69944.4355.057368.5781331.26
171173.371183..6 1013.8376.01382.491 628.86
181219.931230.371026.4401.777429.131329.52
191266.361277.261054422.638429.13 642.6
201312.931324.02115.4448.525462.1031343.68
2113.59371370.91131.8469.357475.726 622.14
221405.931417.661143.8495.241508.851336.54
23----515.968522.395 625.62
24----541.985555.611337.5
25----562.68569.009 619.74
26----588.7602.3511339.06
27----609.273615.705 625.92
28----635.441649.1031339.72
29----655.978662.335 621.42
30----682.152695.451317.88
31----702.554709.056 630.12
32----728.89742.5861341.76
33----749.251755.7 626.94
34----775.6789.3211343.26
35----795.811802.441 637.98
36----822.335836.061343.5
37----842.502849.102 636
38----869.042882.7951345.18
39----889.048895.865 649.02
40----915.775929.5281345.18
41----953.733942.536 648.18
42----962.48976.2611346.86
43----982.266989.316 733
44----1009.211022.991346.8
45----1028.951036 73
46----1055.911069.721348.6
47----1075.471082.79 719.2
48----1102.641116.441348
49----1122.151129.84 719.8
50----1149.341163.181350.4
51----1168.661176.29 737.8
52----1196.071209.891349.2
53----1215.341222.99 739
54----1242.761256.631352.2

Communicated by Elbaz I. Abouelmagd

A0=a12a2284+2e12+2e22+18e14+18e24+23e12e228(γ12+γ22)+8(γ14+γ24)+40γ12γ224(γ12+γ22)e12+e22+Re2a12(1+32e12)+a22(1+32e22)
A1=a12a223e1+14e13+52e1e22+Re2a124e138e1
A2=a12a2212e12+34e14+134e12e22+e12(γ12+γ22)+Re2126a12e12+5a12e14
A3=a1245a22e13Re2e13A4=a12e147164a22+16Re2A5=a12a223e2+14e23+52e12e2+Re2a224e238e2
A6=a12a2212e22+34e24+134e12e22+e22(γ12+γ22)+Re2a22126e22+5e24
A7=a2245a12e23Re2e23A8=a22e247164a12+16Re2
A9=a12a222e1e2158e13e2158e1e23+2e1e2γ12+γ22A10=54a12a22e1e22
A11=198a12a22e1e23,A12=54a12a22e12e2,A13=198a12a22e13e2
A14=10132a12a22e12e22,A15=a12a224e1e2+18e13e218e1e233e1e2γ12+γ22
A16=54a12a22e12e2,A17=138a12a22e13e2,A18=74a12a22e1e22,A19=932a12a22e12e22
A20=98a12a22e1e23,A21=18a12a22e12γ12,A22=98a12a22e22γ22,A23=2a12a22e1γ12
A24=a12a22γ12+12e12γ12+12e22γ12γ145γ12γ22,A25=38a12a22e12γ12,
A26=38a12a22e22γ12,A27=2a12a22e1e2γ12,A28=3a12a22e1e2γ12,
A29=a12a22[2e2γ1298e22γ12],A30=27164a12a22e12e22,A31=92a12a22e1e12,
A32=a12a2232e22+198e12e22+516e24+98γ12e22+98γ22e22,A33=74a12a22e1e12,
A34=564a12a22e12e22,A35=11128a12a22e24,A36=2716a12a22e1e13,A37=116a12a22e1e13,
A38=a12a22e1e22,A39=14a12a22e1e22,A40=1964a12a22e12e22
A41=a12a2234e1e2212e12e22916e24+38e22γ12+38e22γ22,
A42=a12a2212e1e2+12e13e21116e1e23+2e1e2γ12+2e1e2γ22
A43=a12a2212e2+14e12e2+34e23,A44=a12a2212e1e2+3116e13e21916e1e23
A45=14a12a22e12e2,A46=14a12a22e13e2,
A47=a12a2212+34e12764e14+34e22+e12e22764e24+12γ12+12γ22
A48=a12a2212e1+14e1e22+34e13,A49=14a12a22e13,A50=14a12a22e14,
A51=a12a2234e12916e1412e12e22+38e12γ12+38e12γ22,
A52=a12a2292e1e2+e13e2+e1e23+4e1e2γ12+4e1e2γ22,
A53=a12a22+32e2114e12e2e232e2γ122e2γ22,
A54=a12a2232e1e2+78e13e2+136e1e23,A55=a12a22e12e2,A56=78a12a22e13e2,
A57=2316a12a22e1e23,A58=12a12a22e23,A59=1716a12a22e1e23,A60=3128a12a22e24,
A61=98a12a22γ12e22+γ22e12,A62=38a12a22e12γ12,A63=4Re2a1a2γ1γ2σ1σ2
A64=4R2a1a2γ1γ2σ1σ2,A65=92a12a22e12e2,A66=118a12a22e13e2,
A67=18a12a22e13e2,A68=58a12a22e13,A69=3128a12a22e14A70=94a12a22e12e2,
A71=13364a12a22e12e22,A72=2a12a22e2γ12,A73=4a12a22e1e2γ12,
A74=a12a2232e12+516e14458e12e22+98e12γ12+98e12γ22
A75=a12a22γ1212e12γ1212e22γ12+γ14+γ12γ22A76=2a12a22e1γ12,
A77=98a12a22e12γ12,A78=2a12a22e1e2γ12,A79=38a12a22e22γ12
A80=12a12a22γ24,A81=2a12a22γ1γ23,A82=2a12a22γ1γ23,A83=2a12a22e1γ22
A84=a12a22γ22+γ12γ22+γ2412e12γ2212e22γ22,A85=38a12a22e12γ22,
A86=38a12a22e22γ22,A87=2a12a22e1e2γ22,A88=4a12a22e1e2γ22,
A89=2a12a22e2γ22,A90=2a12a22e2γ22,A91=4a12a22e1e2γ22,
A92=a12a22γ22+5γ12γ22+γ2412e12γ2212e22γ22,A93=2a12a22e1γ22
A94=98a12a22e12γ22,A95=2a12a22e1e2γ22,A96=38a12a22e22γ22
A97=3a12a22γ12γ22,A98=94a12a22e12γ1γ2,A99=4a12a22e1γ1γ2,
A100=a12a22γ1γ22+e12+e222γ124γ22A101=34a12a22e12γ1γ2,
A102=34a12a22e22γ1γ2,A103=4a12a22e1e2γ1γ2,A104=8a12a22e1e2γ1γ2,
A105=4a12a22e2γ1γ2,A106=94a12a22e22γ1γ2,
A107=a12a22γ1γ22e12e22+4γ12+4γ22,A108=34a12a22e12γ1γ2,
A109=5a12a22e2γ1γ2,A110=94a12a22e22γ1γ2,A111=92Re2a1a2e1e2γ22,
A112=Re2a1a23e232e2e126e2γ12,A113=Re2a1a232e1e298e13e232e1e2γ12,
A114=98Re2a1a2e12e2,A115=23Re2a1a2e24,
A116=Re2a1a234e22+38e12e22+34e24+34e22γ12+34e22γ22
A117=38Re2a1a2e1e22,A118=98Re2a1a2e12e22,
A119=Re2a1a2e2+12e12e234e23+e2γ12+e2γ22,A120=38Re2a1a2e12e2
A121=Re2a1a212e1e2+38e13e2+38e1e23+12e1e2γ12+12e1e2γ22,
A122=Re2a1a22+2e12+2e22+116e14+116e2412e12e22+4γ12+4γ224γ12γ22,
A123=Re2a1a2e1+34e13+12e1e22+e1γ12+e1γ22
A124=R22a1a234e12+34e14+38e12e22+34e12γ12+34e12γ22,A125=23Re2a1a2e14
A126=Re2a1a214e22+18e12e22112e24+14e22γ122γ22+14e12γ22+54e22γ22+2γ12γ22
A127=18Re2a1a2e1e22,A128=332Re2a1a2e12e22,A129=18Re2a1a2e1e23
A130=112Re2a1a2e23,A131=124Re2a1a2e1e23,A132=364Re2a1a2e24,
A133=92Re2a1a2e1e2γ22,A134=3Re2a1a2e2γ22,A135=32Re2a1a2e1e2γ22
A136=14Re2a1a2e22γ22,A137=Re2a1a2e1γ22,A138=3Re2a1a2e1γ22
A139=34Re2a1a2e12γ22,A140=32Re2a1a2e1e2γ22,A141=Re2a1a2e2γ22,
A142=12Re2a1a2e1e2γ22,A143=34Re2a1a2e22γ22,A144=38Re2a1a2e12e2
A145=Re2a1a214e12γ1214e12γ222γ12γ22,A146=132Re2a1a2e12e22
A147=Re2a1a23e132e1e223e1γ123e1γ22,
A148=Re2a1a214e12+18e12e22112e14+14e12γ12+14e12γ22,A149=112Re2a1a2e13,
A150=364Re2a1a2e14,A151=Re2a1a232e1e2+18e13e298e1e2332e1e2γ1232e1e2γ22,
A152=18Re2a1a2e12e2,A153=124Re2a1a2e13e2,A154=98Re2a1a2e12e2,
A155=332Re2a1a2e12e22,A156=92Re2a1a2e1e2γ12,A157=3Re2a1a2e2γ12,
A158=32Re2a1a2e1e2γ12,A159=3Re2a1a2e1γ12,
A160=34Re2a1a2γ12e22,A161=Re2a1a2e1γ12,A162=Re2a1a2e2γ12
A163=Re2a1a2e1e2γ12,A164=Re2a1a22γ12+e12γ12+e22γ12+2γ12γ22

References

  • [1]

    Escobal,P.R ,October 1963, "Rise and set Times of a Satellite about an Oblate Planet," AIAA Journal Vol. I,No. 10, pp.2305-2310.

  • [2]

    Escobal,P.R. 1965,"Methods of orbit determination, Robert E. Krieger Publishing Company, Huntington, New York, 1965, pp. 26-29 and 360 - 369.

  • [3]

    Lawton, J. A. January-February, 1987."Numerical Method for Rapidly Determining Satellite - Satellite and Satellite-Ground Station In - View Periods, " Journal of guidance, Navigation, and Control, vol. 10,pp.32-36.

  • [4]

    Salvatore Alfano , David Negron, Jr., and Jennifer L. Moore, April-June 1992, "Rapid Determination of Satellite Visibility Periods", The Journal of the Astronautical Sciences, Vol.40, No.2, pp.281-295.

  • [5]

    P. L. Palmer & Yan Mai "A Fast Prediction Algorithm of Satellite Passes", 14th Annual AIAA/USU Conference on Small Satellite, Surrey Space Centre University of Surrey,Guildford,GU2 7XH, UK

  • [6]

    Brouwer, D. and Clemence, G. M.: 1955, Methods of Celestial Mechanics, Academic Press, New York.

  • [7]

    Goldstein, H. & Poole, C. & Safko, J , Celestial Mechanics, Barcelona University

  • [8]

    Michel Capderou, 2005, Satellites Orbits and Missions, Springer.

[1]

Escobal,P.R ,October 1963, "Rise and set Times of a Satellite about an Oblate Planet," AIAA Journal Vol. I,No. 10, pp.2305-2310.

[2]

Escobal,P.R. 1965,"Methods of orbit determination, Robert E. Krieger Publishing Company, Huntington, New York, 1965, pp. 26-29 and 360 - 369.

[3]

Lawton, J. A. January-February, 1987."Numerical Method for Rapidly Determining Satellite - Satellite and Satellite-Ground Station In - View Periods, " Journal of guidance, Navigation, and Control, vol. 10,pp.32-36.

[4]

Salvatore Alfano , David Negron, Jr., and Jennifer L. Moore, April-June 1992, "Rapid Determination of Satellite Visibility Periods", The Journal of the Astronautical Sciences, Vol.40, No.2, pp.281-295.

[5]

P. L. Palmer & Yan Mai "A Fast Prediction Algorithm of Satellite Passes", 14th Annual AIAA/USU Conference on Small Satellite, Surrey Space Centre University of Surrey,Guildford,GU2 7XH, UK

[6]

Brouwer, D. and Clemence, G. M.: 1955, Methods of Celestial Mechanics, Academic Press, New York.

[7]

Goldstein, H. & Poole, C. & Safko, J , Celestial Mechanics, Barcelona University

[8]

Michel Capderou, 2005, Satellites Orbits and Missions, Springer.

Journal Information

Figures

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    Geometry of Satellites Visibility

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    Visibility Intervals Between AQUA and ARIRANG2 during 24-H

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    Visibility Intervals Between AQUA and ARIRANG2 24-H with Oblateness Force

  • View in gallery

    Visibility Intervals Between HST and ODIN For 24-H

  • View in gallery

    Visibility Intervals Between HST and ODIN For 24-H with Oblateness Force

  • View in gallery

    Visibility Intervals Between CFESAT and MTI For 24-H

  • View in gallery

    Visibility Intervals Between CFESAT and MTI For 24-H with Oblateness Force

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