Visual Contact between Two Earth’s Satellites

Problem statement: In this study, an analytical expression for predict ing mutually visible between Earth’s satellites is developed. Approach: A relative satellite-satellite visibility function is developed in terms of orbital elements for two sate llites in the same orbital plane. Also, a simple computational algorithm is expressed to compute the visibility function. Results: This study presents a new and easy method to determine the visibility tim es between two satellites which are in the same orbital plane. Conclusion: The satellite-satellite visibility function is sol ved by an easy treatment.


INTRODUCTION
Mutually visible satellites are defined as two artificial satellites that can maintain direct line of sight between each other for a certain time. Actually, the analysis to be developed is primarily concerned with the rise and set time of a given satellite with respect to another, that is, the time of loss or gain of direct line of sight.
The desirability of having a simple analytic method for predicting when two or more satellites are mutually visible to each other is important for examples in (satellite communications, space warfare).
Escobal proposed a faster method to solve the satellite versus Earth-fixed tracking station problem by developing a closed-form solution for the visibility periods about an oblate Earth. He transformed the satellite and tracking station geometry into a single transcendental equation as a function of eccentric anomaly (Escobal, 1964). Roth developed a method to determine the rise-set time and maximum elevation of a satellite (Roth, 1976). Lawton developed a 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 with a Fourier series. He determined the local periodicity of this curve and then used a numerical search to locate rise-set times (Lawton, 1986). Alfano presented a numerical method to rapidly determine the rise-set times of satellite-satellite and satellite-ground station visibility periods, with the line-of-sight corrected for an oblate Earth. They calculated the problem by using an algorithm which used a space curve modeling technique known as parabolic blending to construct the waveform of the visibility function versus time. The algorithm was worked for all orbital eccentricities and perturbed satellite motion and was provided the visibility function to not become discontinuous as it could from thrusting or deploying a tether (Alfano et al., 1992).
In this study, we develop an analytical expression for predicting the visibility function when two satellites are mutually visible to each other. Also, this visibility function will reduce to a function in terms of the orbital elements when the two satellites are in the same plane. , we express a computational algorithm to compute this visibility function.

Nonvisibility using the visibility function (R V ):
The visibility function defined by Eq. 8 can be used to predict explicitly whether or not satellites are visible to one another. The sign of R V associated with visibility can be obtained by constructing a case in which direct line-of-sight visibility is impossibility as shown in Fig.  2. For this case we have:

Reduction of the visibility function using the orbital elements:
In terms of the orbital eccentricity e, semiparameter p and true anomaly f, the equation of each orbit can be expressed by the relationship: When the two satellites are in the same orbital plane: In this case, Fig. 3 shows when the two satellites are in the same orbital plane, then the two satellites has the same inclination and longitude of ascending node. By using Eq. 9: 1 2 2 1 1 2 1 1 2 2 p p cos (f f ) r . r (1 e cos f )(1 e cos f ) Substituting from Eq. 9 and 10 into Eq. 8: Where: at the given time t.
• The algorithm is completed

RESULTS AND DISCUSSION
A MATHEMATICA 8 programing language was designed and executed for calculating and plotting curves to test the behaviour of the sign of the visibility function R V from Eq. 11 applied for two satellites in the same orbit (have the same orbital elements except the mean anomaly) in two cases: • If the difference in mean anomaly is 10°, logically the two satellites will be in a direct line-of-sight communication case all the time • If the difference in mean anomaly is 180°, logically the two satellites will be in a nonvisibility case all the time The real application on a geostationary satellites shows that: the previouse assumption was done in fact (see Fig. 4 and 5).

CONCLUSION
This study presented a method to rapidly determine the visibility of satellite-satellite visibility periods. This study calculated the problem by using an algorithm which used a simple treatment to construct the visibility function versus time. This algorithm was work for two satellites are in the same orbital plane.
Future work: Now, we are constructing a simple treatment to calculate the visibility function for two earth's satellites in different orbital planes.