## 2 System Model for Laser Time Transfer in Common-View
### Models for Optimized Satellite Orbits Design in Multi-Station Common View  
The two-body problem stands as the predominant model for analyzing spaceship dynamics. Essentially, it revolves around the dynamics of two objects, treating them as point masses and describing the gravitational force between them under the assumption that neither object experiences any external force. The equation of motion for a satellite within the two-body problem is expressed using the gravitational formula as follows [24]:  
\[\overset{\rightharpoonup}{r}=-GM\frac{\overset{\rightharpoonup}{r}}{r^{3}}, \tag{3}\]  
where \(GM\) denotes the Earth's gravitational parameter, \(\overset{\rightharpoonup}{r}\) represents the acceleration vector of the satellite's motion, \(\overset{\rightharpoonup}{r}\) is the position vector of the satellite in the specified coordinate system, and \(r=|\overset{\rightharpoonup}{r}|\) indicates the radius. The state vector in the two-body problem can be represented by the position and velocity vectors (\(\overset{\rightharpoonup}{r},\overset{\rightharpoonup}{r}\)) or the Keplerian orbital sextuplets: the semimajor axis (\(a\)), the numerical eccentricity (\(e\)), the inclination (\(i\)), the right ascension of the ascending node (\(\Omega\)), the argument of perigee (\(\omega\)), and the mean anomaly (\(M\)). During the orbital motion of satellites, there are various additional and complex perturbing forces at play. These include the Earth's gravitational field, as well as gravitational influences from the Sun and Moon, Earth's solid tides, ocean tides, and other forces that are not yet fully comprehended. While these forces are much smaller in magnitude compared to the Earth's gravitational force, they still have an impact on the satellite's orbit [25]. To obtain the satellite's in-orbit state vectors, we can utilize the six known orbital parameters and convert them into the satellite's instantaneous position vector. The unit vector in the direction of the perigee is denoted as \(P\), while \(Q\) represents the unit vector oriented perpendicular to the perigee within the orbital plane, aligned with the satellite's orbital motion. \(E\) is the eccentricity anomaly, \(a\) and \(e\) are the semimajor axis and eccentricity. The satellite's instantaneous position vector can be expressed in terms of its orbit direction as follows:  
\[R_{SCTS}= \begin{bmatrix}X_{SCTS}\\ Y_{SCTS}\\ Z_{SCTS}\end{bmatrix} =a(\cos E-e) \begin{bmatrix}P_{x}\\ P_{y}\\ P_{z}\end{bmatrix} +a\sqrt{1-e^{2}}\text{sin}E \begin{bmatrix}P_{x}\\ P_{y}\\ P_{z}\end{bmatrix} , \tag{4}\]  
Coordinates from the satellite's celestial coordinate system can be translated to the instantaneous Earth coordinate system by inputting the instantaneous Greenwich accurate sidereal time (\(GAST\)) at a given moment. It is possible to estimate the Greenwich true sidereal time using the Greenwich mean sidereal time (\(GMST\)):  
\[GMST=GMST0+T_{r}[UTC+(UT1-UTC)]=GMST0+T_{r}UT1, \tag{5}\]  
where \(GMST0\) refers to the Greenwich Mean Time at midnight of the relevant day, while \(T_{r}\) refers to the rate of change and \(UT1\) denotes the Universal Time that has been corrected for pole shift. \(GMST0\) and \(T_{r}\) are determined by the time that has elapsed in the Julian calendar element from J2000.0 to zero \(UT1\) on the day when the measurement is taken, according to empirical formulas. \[R_{S}= \begin{bmatrix}X_{S}\\ Y_{S}\\ Z_{S}\end{bmatrix} = \begin{bmatrix}\cos\ (GMST)&\sin\ (GMST)&0\\ -\sin\ (GMST)&\cos\ (GMST)&0\\ 0&0&1\end{bmatrix} R_{SCTS}, \tag{6}\]  
The satellite's current Earth coordinate system undergoes a conversion process into the Earth geodetic coordinate system. This enables us to acquire the corresponding longitude \(L_{S}\), latitude \(B_{S}\), and height \(H_{S}\). These coordinates can then be inverted to the Earth coordinate system coordinates of the established station, \(L_{r}\) station-centered coordinate system accommodation, using the Earth geodetic coordinate system of the satellite. \[R_{Sr}=