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Calculation of line of site periods between two artificial satellites under the action air drag

M. K. Ammar
M. K. Ammar
, 
M. R. Amin
M. R. Amin
 e 
M.H.M. Hassan
M.H.M. Hassan
   | 01 lug 2018
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Pubblicato online: 01 lug 2018
Pagine: 339 - 352
Ricevuto: 23 gen 2018
Accettato: 20 giu 2018
DOI: https://doi.org/10.21042/AMNS.2018.2.00027
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© 2018 M. K. Ammar et al., published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.
Introduction

Rise–set time computation through the accurate orbit estimation is very important because it plays an essential role in the pre-request information for mission analysis and on-board resources management in many general communication, scientific spacecraft and Earth observation. Also, to provide and exchange information for a wide range of military and civil applications such as communications, there is a big trend to use fast access, low cost and multifunctional small satellites. This requires accurate estimation of when the satellites disappears from the horizon (set) over a time-scale of months in some cases and when the satellite will start to be visible (rise) to a given location on the Earth or to other satellite. Therefore, we referred in Ammar and Hassan [1], to

the rise/set problem which may be defined as the process of determining the times at which a satellite rises and sets with respect to a ground location. The numerical method is the easiest solution to determine the 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. Computation time is a drawback to this method, especially when modeling many perturbations and processing several satellites. Escobal [2], [3] 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 [4] 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 [5] 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 [6], 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, we established a fast way for satellite-satellite visibility intervals for the rise-and-set time prediction for two satellites in terms of classical orbital elements of the two satellites and time. We have considered the secular variations of the orbital elements due to air drag force in order to determine 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.

Visibility Analysis

In order to fully describe the position of a satellite in space at any given time, we used a set of six orbital parameters semi-major axis a, eccentricity e, inclinations of the orbit plane i, right ascension of the node Ω, the argument of perigee ω, and true anomaly f. The above parameters are shown in the Fig.1.

Fig. 1

Geometry of Satellites Visibility

The visibility function, U, which describes whether these two satellites can achieve visibility were derived in Ammar and Hassan [1], Eq. 1 and in briefly it can be obtained as follows:

U=Re2r12+r22−2r→1⋅r→2−r12r22+r→1⋅r→22$$\begin{equation}\label{1} U = R_e^2\left[ {\left( {{r_1}^2 + {r_2}^2} \right) - 2\left( {{{\vec r}_1} \cdot {{\vec r}_2}} \right)} \right] - {r_1}^2{r_2}^2 + {\left( {{{\vec r}_1} \cdot {{\vec r}_2}} \right)^2} \end{equation}$$

Where

U=+ve,Non−visibilitycase0,riseorset−ve,direct−lineof−sight$$U = \left\{ {\begin{array}{*{20}{c}}{ + ve\,\,,\,Non - visibility\,case\,}\\{0\,\,\,\,,\,\,rise\,\,or\,\,set \qquad}\\{ - ve\,\,,\,\,direct - \,line\,of - \,sight\,}\end{array}} \right.$$

Referring to Fig.1, the position vectors of satellites 1, and 2 with respect to the ECI coordinate system arer→1.$\vec r_1.$andr→2.$\vec r_2.$

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

Construction of The Visibility Function

The position vector of each satellite in the geocentric coordinate system, r→=(x,y,z),$\vec r = (x,y,z)$can be calculated by the following formula [7],

xyz=rcos⁡Ωcosω+f−sin⁡Ωsinω+fcos⁡isin⁡Ωcosω+f+cos⁡Ωsinω+fcos⁡isinω+fsin⁡i$$\begin{equation*}\left( {\begin{array}{*{20}{c}}x\\y\\z\end{array}} \right) = r\left( {\begin{array}{*{20}{c}}{\cos \Omega \cos \left( {\omega + f} \right) - \sin \Omega \sin \left( {\omega + f} \right)\cos i}\\{\sin \Omega \cos \left( {\omega + f} \right) + \cos \Omega \sin \left( {\omega + f} \right)\cos i}\\{\sin \left( {\omega + {f}} \right)\sin i}\end{array}} \right) \end{equation*}$$

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

r=a(1−e2)1+ecos⁡f$$\begin{equation*}r = \frac{{a(1 - {e^2})}}{{1 + e\cos f}} \end{equation*}$$

Forming scalar product r→1⋅r→2,$\left( {{{\vec r}_1} \cdot {{\vec r}_2}} \right),$keeping terms up to O(e4) only, we obtain

r→1⋅r→2=x1x2+y1y2+z1z2$$\begin{equation*}{\vec r_1} \cdot {\vec r_2} = {x_1}\,{x_2} + {y_1}\,{y_2} + {z_1}\,{z_2} \end{equation*}$$

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

x1=r1σ12cos⁡(f1+ω1+Ω1)+γ12cos⁡(f1+ω1−Ω1)y1=r1σ12sin⁡(f1+ω1+Ω1)−γ12sin⁡(f1+ω1−Ω1)z1=2r1σ1γ1cos⁡(f1+ω1)$$\begin{equation}\begin{array}{l}{x_1} = {r_1}\left[ {{\sigma _1}^2\cos ({f_1} + {\omega _1} + {\Omega _1}) + {\gamma _1}^2\cos ({f_1} + {\omega _1} - {\Omega _1})} \right]\\\\{y_1} = {r_1}\left[ {{\sigma _1}^2\sin ({f_1} + {\omega _1} + {\Omega _1}) - {\gamma _1}^2\sin ({f_1} + {\omega _1} - {\Omega _1})} \right]\\\\{z_1} = 2\,{r_1}\,{\sigma _1}\,{\gamma _1}\cos ({f_1} + {\omega _1})\end{array} \end{equation}$$

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 [7] up to O(e4),

r1=a11+12e12+−e1+38e13cosM1+−12e12+13e14cos⁡2M1−38e13cos⁡3M1−13e14cos⁡4M1$$\begin{equation}\begin{array}{rl}{r_1} =& {a_1}\left[ {\left( {1 + \frac{1}{2}e_1^2} \right)} \right. + \left( { - {e_1} + \frac{3}{8}e_1^3} \right)\cos {M_1} + \left( { - \frac{1}{2}e_1^2 + \frac{1}{3}e_1^4} \right)\cos 2{M_1}\,\\\\ &- \frac{3}{8}e_1^3\cos 3{M_1} - \left. {\frac{1}{3}e_1^4\cos 4{M_1}} \right]\end{array} \end{equation}$$r1cosf1=a1−32e+1−38e12+5192e14cosM1−12e1+13e13cos⁡2M1+−12e12+13e14cos⁡2M1+38e12−45128e13cos⁡3M1+13e13cos⁡4M1$$\begin{equation}\begin{array}{rl} {r_1}\cos {f_1} =& {a_1}\left[ { - \frac{3}{2}e + \left( {1 - \frac{3}{8}e_1^2 + \frac{5}{{192}}e_1^4} \right)} \right.\cos {M_1} - \left( {\frac{1}{2}{e_1} + \frac{1}{3}e_1^3} \right)\cos 2{M_1}\\\\ & + \left( { - \frac{1}{2}e_1^2 + \frac{1}{3}e_1^4} \right)\cos 2{M_1} + \left( {\frac{3}{8}e_1^2 - \frac{{45}}{{128}}e_1^3} \right)\cos 3{M_1} + \left. {\frac{1}{3}e_1^3\cos 4{M_1}} \right]\end{array} \end{equation}$$r1sinf1=a11−58e12−11192e14sinM1+12e1−512e13sin⁡2M1+38e12−51128e14sin⁡3M1+13e13sin⁡4M1$$\begin{equation}\begin{array}{rl} {r_1}\sin {f_1} =& {a_1}\left[ {\left( {1 - \frac{5}{8}e_1^2 - \frac{{11}}{{192}}e_1^4} \right)} \right.\sin {M_1} + \left( {\frac{1}{2}{e_1} - \frac{5}{{12}}e_1^3} \right)\sin 2{M_1}\\\\ & + \left( {\frac{3}{8}e_1^2 - \frac{{51}}{{128}}e_1^4} \right)\sin 3{M_1} + \left. {\frac{1}{3}e_1^3\sin 4{M_1}} \right]\end{array} \end{equation}$$

With similar expressions for the other satellite. Substituting Eqs. ( 3–5 ) into Eq. (2), and keeping terms up to O(e4), we obtain the visibility function [Ammar and Hassan [1]]:

The Effect of Drag

The acceleration due to air drag has the general form [8]

F→D=−12mCDAsρairV2V^_$$\begin{equation*}{\vec F_D} = - \frac{1}{{2m}}{C_D}{A_s}\,{\rho _{air}}\,{V^2}\,\;\underline {\hat {V}} \end{equation*}$$

Where, m is the satellite mass, C D is the aerodynamic drag coefficient, As is the average cross-sectional area of the satellite, rair is the air density and V is the magnitude of the satellite velocity relative to the atmosphere, and is the unit vector in the satellite velocity direction.

Since the drag force is non-conservative, so we will use Lagrange’s planetary equations in Gaussian form Roy [9] expressed in the R SW- coordinate system, i.e. in the directions of the radial, transverse and orthogonal respectively, shown in Fig. 2.

Fig. 2

The relation between RSW and TNW-Coordinate systems

Also, since the drag force in the opposite direction of the velocity vector, then we can express the drag acceleration components in the form:

F→D=FR,FS,0=−FDcos⁡φ,−FDsin⁡φ,0$$\begin{equation*}\label{10} {\vec F_D} = \left( {{F_R},{F_S},0} \right) = \left( { - \left| {{F_D}} \right|\cos \varphi , - \left| {{F_D}} \right|\sin \varphi \,\,,0} \right) \end{equation*}$$

Where φ is the flight path angle.

Expressing sinφ and cosφ in terms of the true anomaly f, by using the well-known relations:

cosφ=1+ecos⁡f1+ecos⁡f+e2,sinφ=esin⁡f1+ecos⁡f+e2$$\begin{equation*}\cos \,\varphi = \frac{{1 + e\cos f}}{{\sqrt {1 + e\cos f + {e^2}} }} ,\,\,\,\,\, \sin \,\varphi = \frac{{e\sin f}}{{\sqrt {1 + e\cos f + {e^2}} }} \end{equation*}$$

We can write the rate of change of the osculating elements of the satellite in the RSW- Coordinate system in the form :

dadt=2n1−e2esin⁡fFR+1+ecos⁡fFS$$\begin{equation*}\frac{{da}}{{dt}} = \frac{2}{{n\sqrt {1 - {e^2}} }}\left[ {\left( {e\sin f} \right){F_R} + \left( {1 + e\cos f} \right){F_S}} \right] \end{equation*}$$dedt=1−e2nasin⁡fFR+e+2cos⁡f+ecos2f1+ecos⁡fFS$$\begin{equation*}\frac{{de}}{{dt}} = \frac{{\sqrt {1 - {e^2}} }}{{na}}\left[ {\left( {\sin f} \right)\,{F_R} + \frac{{\left( {e + 2\cos f + e{{\cos }^2}f} \right)}}{{\left( {1 + e\cos f} \right)}}{F_S}} \right] \end{equation*}$$dωdt=1−e2nae−cos⁡fFR+2+ecos⁡f1+ecos⁡f(sin⁡f)FS$$\begin{equation*}\frac{{d\omega }}{{dt}} = \frac{{\sqrt {1 - {e^2}} }}{{nae}}\left[ { - \left( {\cos f} \right){F_R} + \frac{{\left( {2 + e\cos f} \right)}}{{\left( {1 + e\cos f} \right)}}(\sin f){F_S}} \right] \end{equation*}$$dMdt=n+1−e2naesin⁡f−2e1+ecos⁡fFR−2+ecos⁡f1+ecos⁡fsin⁡fFS$$\begin{equation*}\frac{{dM}}{{dt}} = n + \frac{{\left( {1 - {e^2}} \right)}}{{nae}}\left[ {\left( {\sin f - \frac{{2e}}{{\left( {1 + e\cos f} \right)}}} \right){F_R} - \frac{{\left( {2 + e\cos f} \right)}}{{\left( {1 + e\cos f} \right)}}\left( {\sin f} \right){F_S}} \right] \end{equation*}$$

Since the drag force oppose the velocity vector. Hence, we need to find the drag components in the TNW - coordinate system, where T- axis aligned along the tangent (velocity vector), N - axis normal to it in the direction of increasing the true anomaly, f, and W - axis completes the triad in the positive sense. The relations between the two systems are given from Fig.2, after eliminating the flight path angle φ, between them as:

FR=1+ecos⁡f1+e2+2ecos⁡fFT−esin⁡f1+e2+2ecos⁡fFNFS=esin⁡f1+e2+2ecos⁡fFT+1+ecos⁡f1+e2+2ecos⁡fFN$$\begin{equation*}\begin{array}{rl}{F_R} = \frac{{\left( {1 + e\cos f} \right)}}{{\sqrt {1 + {e^2} + 2e\cos f} }}\,{F_T} - \frac{{\left( {e\sin f} \right)}}{{\sqrt {1 + {e^2} + 2e\cos f} }}\,{F_N}\\\\\\ {F_S} = \frac{{\left( {e\sin f} \right)}}{{\sqrt {1 + {e^2} + 2e\cos f} }}\,{F_T} + \frac{{\left( {1 + e\cos f} \right)}}{{\sqrt {1 + {e^2} + 2e\cos f} }}\,{F_N}\end{array} \end{equation*}$$dadtD=−BCρVs2n1−e21+e2+2ecos⁡fdedtD=−BCρVs21−e2nae+cos⁡f1+e2+2ecos⁡fdωdtD=−BCρVs21−e2naesin⁡f1+e2+2ecos⁡fdMdtD=n−BCρVs2nae1−e2sin⁡f1+e2+2ecos⁡f11−e2+1−e2−11+ecos⁡f$$\begin{equation}\begin{array}{rl} & {\left( {\frac{{da}}{{dt}}} \right)_D} = - \frac{{{B_C}\,\rho \,V_s^2}}{{n\sqrt {1 - {e^2}} }}\sqrt {1 + {e^2} + 2e\cos f} \\\\ & {\left( {\frac{{de}}{{dt}}} \right)_D} = - \frac{{{B_C}\,\rho \,V_s^2\,\sqrt {1 - {e^2}} }}{{n\,a}}\frac{{\left( {e + \cos f} \right)}}{{\sqrt {1 + {e^2} + 2e\cos f} }}\\\\ & {\left( {\frac{{d\omega }}{{dt}}} \right)_D} = - \frac{{{B_C}\,\rho \,V_s^2\sqrt {1 - {e^2}} }}{{n\,a\,e}}\frac{{\sin f}}{{\sqrt {1 + {e^2} + 2e\cos f} }}\\\\ & {\left( {\frac{{dM}}{{dt}}} \right)_D} = n - \frac{{{B_C}\,\rho \,V_s^2}}{{n\,a}}\frac{{e\left( {1 - {e^2}} \right)\sin f}}{{\sqrt {1 + {e^2} + 2e\cos f} }}\left( {\frac{1}{{\left( {1 - {e^2}} \right) + \sqrt {1 - {e^2}} }} - \frac{1}{{1 + e\cos f}}} \right)\end{array} \end{equation}$$

Where, BC=AmCD${B_C} = \frac{A}{m}{C_D}\,$, called the ballistic coefficient.

We have to express the satellite velocity Vs2 in the form:

Vs2=μa1−e21+2ecos⁡f+e2$$\begin{equation}V_s^2 = \frac{\mu }{{a\left( {1 - {e^2}} \right)}}\left( {1 + 2e\cos f + {e^2}} \right) \end{equation}$$

Substituting Eq. 7 into Eq. 6 we have the variations of the orbital elements due to drag in the form:

dadtD=−μBCρna1+2ecos⁡f+e23/21−e23/2dedtD=−μBCρna2e+cos⁡f1−e21+e2+2ecos⁡fdωdtD=−μBCρna2esin⁡f1−e21+e2+2ecos⁡fdMdtD=n−μBCρna2eesin⁡f1−e2+1−e2−11+ecos⁡f1+e2+2ecos⁡f$$\begin{equation*}\label{19} \begin{array}{rl} &{\left( {\frac{{da}}{{dt}}} \right)_D} = - \left( {\frac{{\mu {B_C}\,\rho }}{{n\,a}}} \right)\,\,\frac{{{{\left( {1 + 2e\cos f + {e^2}} \right)}^{3/2}}}}{{{{\left( {1 - {e^2}} \right)}^{3/2}}}}\\\\\\ &{\left( {\frac{{de}}{{dt}}} \right)_D} = - \left( {\frac{{\mu {B_C}\,\rho }}{{n\,{a^2}}}} \right)\left( {\frac{{e + \cos f}}{{\sqrt {1 - {e^2}} }}} \right)\sqrt {1 + {e^2} + 2e\cos f} \\\\\\ &{\left( {\frac{{d\omega }}{{dt}}} \right)_D} = - \left( {\frac{{\mu {B_C}\,\rho }}{{n\,{a^2}e}}} \right)\left( {\frac{{\sin f}}{{\sqrt {1 - {e^2}} }}\sqrt {1 + {e^2} + 2e\cos f} } \right)\\\\\\ &{\left( {\frac{{dM}}{{dt}}} \right)_D} = n - \left( {\frac{{\mu {B_C}\,\rho }}{{n\,{a^2}e}}} \right)\left( {\frac{{e\sin f}}{{\left( {1 - {e^2}} \right) + \sqrt {1 - {e^2}} }} - \frac{1}{{1 + e\cos f}}} \right)\sqrt {1 + {e^2} + 2e\cos f} \end{array} \end{equation*}$$

Since we shall consider only the secular effects of the drag force on the motion of the satellites, we average Eq. 7 with respect to the true anomaly f, to obtain:

dadt¯D=−μBCρna1+34e2+2164e4$$\begin{equation}{\left( {\overline {\frac{{da}}{{dt}}} } \right)_D} = - \left( {\frac{{\mu {B_C}\,\rho }}{{na}}} \right)\left( {1 + \frac{3}{4}{e^2} + \frac{{21}}{{64}}{e^4}} \right) \end{equation}$$dedt¯D=−μBCρna212e−516e3$$\begin{equation}{\left( {\overline {\frac{{de}}{{dt}}} } \right)_D} = - \left( {\frac{{\mu {B_C}\,\rho }}{{n{a^2}}}} \right)\left( {\frac{1}{2}e - \frac{5}{{16}}{e^3}} \right) \end{equation}$$dωdt¯D=0,dMdt¯D=n$$\begin{equation*}{\left( {\overline {\frac{{d\omega }}{{dt}}} } \right)_D} = 0\,\,\,\,,\,\,\,\,\,\,\,\,{\left( {\overline {\frac{{dM}}{{dt}}} } \right)_D} = n \end{equation*}$$

Where the bar indicates that these rates contain secular terms only.

Therefore, the drag cause secular changes only on the semi major axis and the eccentricity of the satellite orbit.

We can now consider the air density p in the form [9] as:

ρ=ρ0e−η−η0−η−η0HH$$\begin{equation*}\rho = {\rho _0}\,{e^{{{ - \left( {\eta - {\eta _0}} \right)} \mathord{\left/ {\vphantom {{ - \left( {\eta - {\eta _0}} \right)} H}} \right.} H}}} \end{equation*}$$

Where

po is the air density at perigee,

η is the satellite altitude,

ηo is the altitude at the perigee,

H is the scale height.

With the relation μ = n2a3 we can rewrite Eqs. 8, 9 as

dadt¯D=−na2Bcρ0e−η−η0−η−η0HH1+34e2+2164e4$$\begin{equation}{\left( {\overline {\frac{{da}}{{dt}}} } \right)_D} = - n\,{a^2}\left( {{B_c}\,{\rho _0}{e^{{{ - \left( {\eta - {\eta _0}} \right)} \mathord{\left/ {\vphantom {{ - \left( {\eta - {\eta _0}} \right)} H}} \right. } H}}}} \right)\left( {1 + \frac{3}{4}{e^2} + \frac{{21}}{{64}}{e^4}} \right) \end{equation}$$dedt¯D=−naBcρ0e−η−η0−η−η0HH12e−516e3$$\begin{equation}{\left( {\overline {\frac{{de}}{{dt}}} } \right)_D} = - n\,a\left( {{B_c}\,{\rho _0}{e^{{{ - \left( {\eta - {\eta _0}} \right)} \mathord{\left/ {\vphantom {{ - \left( {\eta - {\eta _0}} \right)} H}} \right. } H}}}} \right)\left( {\frac{1}{2}e - \frac{5}{{16}}{e^3}} \right) \end{equation}$$

Integrating Eqs. 10 and 11 with respect to the time t we obtain the secular variation in the semi-major axis and eccentricity due to air drag in the form:

Δa¯D=−na2Bcρ0e−η−η0−η−η0HH1+34e2+2164e4t$$\begin{equation*}\Delta {\bar a_D} = - n\,{a^2}\left( {{B_c}\,{\rho _0}{e^{{{ - \left( {\eta - {\eta _0}} \right)} \mathord{\left/ {\vphantom {{ - \left( {\eta - {\eta _0}} \right)} H}} \right. } H}}}} \right)\left( {1 + \frac{3}{4}{e^2} + \frac{{21}}{{64}}{e^4}} \right)\;t\,\, \end{equation*}$$Δe¯D=−naBcρ0e−η−η0−η−η0HH12e−516e3t$$\begin{equation*}\Delta {\bar e_D} = - n\,a\left( {{B_c}\,{\rho _0}\,{e^{{{ - \,\,\left( {\eta - \,{\eta _0}} \right)} \mathord{\left/ {\vphantom {{ - \,\,\left( {\eta - \,{\eta _0}} \right)} H}} \right. } H}}}} \right)\left( {\frac{1}{2}e - \frac{5}{{16}}{e^3}} \right)\,t\, \end{equation*}$$

That represents the secular changes in the orbit due to air drag.

Adding Perturbing Forces

We shall consider the effect of perturbation on the orbital elements due to the atmospheric drag. So, We will express the orbital elements of the two satellites in the form:

σj(t)=σjo+ΔσjDj=1,2.$$\begin{equation*}\label{28} {\sigma _j}(t) = {\sigma _{jo}} + {\left( {\Delta {\sigma _j}} \right)_{D}} \,\,\,\,\,\,\,\,\,\,\,\,\, j =1,2. \end{equation*}$$

Where σj (t) represent respectively any of the orbital elements, σj0 the unperturbed element, and (Δσj)Ddenote the perturbations in the elements due to drag force. The expansion of the perturbed visibility function about some epoch time t0 can be obtained by Taylor expansions about the osculating elements (a0j,e0j, i0j,Ω0j,ω0j,M0j) up to the first order as:

F(aj,ej,ij,Ωj,ωj,Mj)=U(a0j,e0j,i0j,Ω0j,ω0j,M0j)+∑s=16∂U∂σs0Δσs$$\begin{equation}\label{29} F({a_j},{e_j},{i_j},{\Omega _j},{\omega _j},{M_j}) = U({a_{_0j}},{e_{0j}},{i_{0j}},{\Omega _0}_j,{\omega _0}_j,{M_0}_j) + \sum_{s=1}^6 {{{\left( {\frac{{\partial U}}{{\partial {\sigma _s}}}} \right)}_0}} \Delta {\sigma _s} \end{equation}$$

The summation ranges from s = 1 to s = 4, where s = 1 , 2 represent the elements (a1,e1) and s = 3 , 4 represent (a2,e2) respectively.

Numerical Examples

In what follows the visibility function were tested for some examples to obtain the mutual visibility between two Earth Satellites. The orbital elements for some satellites were obtained from the Center for Space Standards & Innovation and are listed in Tables 1, 2.

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
p (kg=km3)3.63E-054.6E-050.0003540.000369
po (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
B*2.5E-053.76E-051.36E-055.61E-05
BC =CDA/m (m2/kg)5.4E-058.1E-056.11E-062.52E-05

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
p (kg=km3)0.0011620.003008
po (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
B*6.71E-054.84E-05
BC =CDA/m (m2/kg)1.33E-054.07E-06

The visibility intervals with the action of air drag are shown in Figures 4, 6, 8 according as the sign of the visibility function given in Eq. (12) and without any perturbing force are shown in Figures 35, 7, and are listed in Table 3, 4 and 5.

Fig. 3

Visibility Intervals Between AQUA and ARIRANG2 during 24-H

Fig. 4

Visibility Intervals Between AQUA and ARIRANG2 24-H with Air Drag Force

Fig. 5

Visibility Intervals Between HST and ODIN For 24-H

Fig. 6

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

Fig. 7

Visibility Intervals Between CFESAT and MTI For 24-H

Fig. 8

Visibility Intervals Between CFESAT and MTI For 24-H Air Drag Force

Visibility Intervals Between AQUA And ARIRANG2 During 24 Houres

Without Air Drag ForceWith Air Drag Force
visibility timevisibility time
RiseSetmsRiseSetms
117.107444.86632745.53417.102744.86762745.894
266.408694.04822738.37666.414594.05422738.382
3115.669143.3182738.94115.674143.3062737.92
4164.973192.4992731.56164.973192.5192732.76
5214.236241.7682731.92214.25241.7422729.52
6263.542290.9482724.36263.536290.9822726.76
7312.807340.2162724.54312.83340.1772720.82
8362.115389.3962716.86362.103389.4432720.4
9411.382438.6632716.86411.416438.612711.64
10460.692487.842278.94460.675487.9032713.68
11509.962537.108278.76510.006537.041272.1
12559.276586.286276559.251586.362276.66
13608.547635.552270.3608.6635.4722652.32
14657.863684.7292651.96657.831684.8192659.28
15707.135733.9942651.54707.199733.92642.06
16756.453783.1712643.08756.416783.2752651.54
17805.728832.4352642.42805.802832.3272631.5
18855.048881.6112633.78855.004881.7292643.5
19904.325930.8742632.94904.41930.7532620.58
20953.647980.0492624.12953.597980.1822635.1
211002.931029.312622.81003.021029.18269.6
221052.251078.492614.41052.191078.632626.4
231101.351127.752613.21101.641127.62557.6
241150.861176.92263.61150.791177.082617.4
251200.141226.18262.41200.261226.022545.6
261249.471275.362553.41249.41275.53267.8
271298.751324.622552.21298.881324.442533.6
281348.081373.792542.61348.011373.892558.2
291397.371423.052540.81397.51422.682521

Visibility Intervals Between HST and ODIN 24 Houres

Without Air Drag ForceWith Air Drag Force
RiseSetvisibility timeRiseSetvisibility time
msms
139.325544.443257.06239.320844.443257.344
287.203592.0758452.33887.16892.1059456.274
3134.82139.91355.52134.901139.846456.7
4182.699187.545450.76182.584187.63553.06
5230.316235.38354.02230.484235.246445.72
6278.195283.014449.14278.002283.16359.66
7325.812330.85252.4326.072330.642434.2
8373.691378.84359.12373.42378.69516.26
9421.307426.32250.9421.663426.033422.2
10469.188473.952445.84468.84474.217522.62
11516.803521.792459.34517.259521.4249.66
12564.684569.421444.22564.261569.743528.92
13612.298617.261457.72612.861616.8356.34
14660.18664.98448659.682665.267535.1
15707.794712.731456.22708.47712.175342.3
16755.676760.359440.98755.105760.791541.16
17803.289808.201454.72804.087807.541327.24
18851.172855.828439.18850.528856.314547.16
19898.785903.67453.1899.714902.896310.92
20964.668951.297437.74945.952951.836553.04
21994.28999.14451.6995.353998.24253.22
221042.161046.77436.61041.381047.36531.8
231089.781094.61449.81091.011093.57233.6
241137.661142.23434.21136.81142.8864.8
251185.271190.08448.61186.691188.87210.8
261233.161237.7432.41232.231238.4610.2
271280.771285.55446.81282.421284.12142
281328.651333.17431.21327.661333.92615.6
291376.261381.02445.61378.231379.2913.6
301424.151428.64429.41423.081429.44621.6

Visibility Intervals Between CFESAT and MTI 24 Houres

Without Air Drag ForceWith Air Drag Force
RiseSetvisibility timeRiseSetvisibility time
msms
1430.519432.975227.36383.281386.757328.56
2476.988479.867252.74475.837480.88853.06
3523.051527.14745.76568.562574.83616.08
4569.583573.973423.4617.63619.378144.88
5615.801621.095517.64661.372668.709720.22
6662.353667.899532.76710.16713.559323.94
7708.642714.947618.3754.234762.517816.98
8755.203761.74632.22802.941807.477432.22
9801.535808.742712.42847.133856.28399
10848.101855.526725.5895.808901.304529.76
11894.462902.49682.04940.061950.015957.24
12941.03949.275814.7988.72995.079621.54
13987.413996.22848.421033.011043.721042.6
141033.981042.9990.61081.661088.8279.6
151080.381089.92932.41125.981137.41125.2
161126.951136.69944.41174.611182.52754.6
171173.371183..61013.81218.971231.06125.4
181219.931230.371026.41267.581276.23839
191266.361277.2610541311.981324.691242.6
201312.931324.02115.41360.551369.91921.6
2113.59371370.91131.814051418.311318.6
221405.931417.661143.8----
Conclusions

We referred to the first column (The Function of Visibility without any perturbation) in Ammar and Hassan [1] of this paper, now we refer to the second column (The Function of Visibility with the Air Drag Force)

In the Table 3 (Visibility Intervals Between AQUA and ARIRANG2 ), In the second column (with the Air Drag Force), the increase and decrease in oscillation is noticeable, the time of large periods increases and the time of the small periods decreases gradually, then the effect of the air drag force appears clearly.

In the Table 4 (Visibility Intervals Between HST and ODIN ), In the second column (with the Air Drag Force), the increase and decrease in oscillation is noticeable, the time of large periods increases and the time of the small periods decreases gradually, then the effect of the air drag force appears clearly.

In Table 5 (Visibility Intervals Between CFESAT and MTI), In the second column (with the Air Drag Force), the increase in oscillation is noticeable and greater than the previous examples, because the semi-major axis is smaller than the other one in the previous examples and less than 600 Km, then the effect of the air drag force appears clearly. It is also noticed that there is a low number of periods of visibility function that affects the air Drag force.

The secular variations of the orbital elements due the Effect of the Air Drag Force was considered and it appeared obviously in the previous tables. The new method exploits sophisticated analytic models of the orbit and therefore provides direct computation of rise-set times. Numerical examples for some satellites were given to chick the validity of the method.

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