## 2 Constellation Station Maintenance Analysis
### Analysis of Satellite Constellation Phase Evolution  
The Earth's non-spherical perturbation terms can be divided into zonal harmonics and tesseral harmonics. Among them, the influence of the first term of the zonal harmonics is much greater than that of other terms. Therefore, this paper only considers the perturbation influence of this term. As a conservative force, the Earth's non-spherical perturbation does not have a long-term perturbation effect on the orbital elements \(a,\;e,i\) of a circular orbit, but will only cause periodic fluctuations [21].  
\[\left\{\begin{array}{l}\dot{\Omega}=-\frac{3\lambda_{2}R_{e}^{2}}{2p^{2}}n \cos i\\ \dot{\lambda}=-\frac{3\lambda_{2}R_{e}^{2}}{2p^{2}}n\Big{[}\big{(}2-\frac{5}{ 2}\sin^{2}i\big{)}-\big{(}1-\frac{3}{2}\sin^{2}i\big{)}\sqrt{1-e^{2}}\Big{]} \end{array}\right., \tag{1}\]  
In the formula, \(\dot{\Omega}\) is the drift rate of the right ascension of the ascending node, \(J_{2}\) is the Earth's non-spherical perturbation term, \(\dot{\lambda}\) is the drift rate of the phase angle, \(R_{e}\) is the radius of the Earth, \(p\) is the semi\(-\)latus rectum, \(n\) is the average angular velocity of the satellite's motion, \(i\) is the orbital inclination, and \(e\) is the eccentricity.  
As the orbital altitude gradually increases, the drift rate of the phase angle shows a significant linear decrease. Therefore, the change in the satellite's orbital altitude plays a major role in the process of the phase angle drift.  
Ideally, for low-orbit satellites in near-circular orbits, the nominal orbital semi-major axes and orbital inclinations of satellites within the same orbital plane of the constellation should be consistent. If only the perturbations of the Earth's non-spherical J2 term and atmospheric drag are considered, the perturbation and drift laws of the orbital elements of low-orbit constellation satellites are also consistent. However, for satellites in different orbital planes of the constellation, under the influence of the Earth's non-spherical J2 term and atmospheric drag, the attenuation of the satellite's semi-major axis and the laws of the phase change are inconsistent, so they need to be discussed separately. In view of this, the subsequent analysis of the evolution of the satellite phase is applicable only to the satellites within the same orbital plane of the low-orbit satellite constellation in near-circular orbits.  
For low-orbit satellites in near-circular orbit, the attenuation of the orbital semi-major axis caused by atmospheric drag will cause a relative phase drift of the satellite. The relationship between the phase angle drift rate and the semi-major axis attenuation can be written as follows [15]:  
\[\Delta\dot{\lambda}(t)=-\frac{3}{2}\sqrt{\mu}a^{-\frac{5}{2}}\Delta a, \tag{2}\]  
By integrating the above formula, the relative rate of change in the phase angle can be regarded as a function of the semi-major axis and the rate of change of the semi-major axis. The relationship between the phase angle drift and the semi-major axis attenuation is [15].  
\[\Delta\lambda(t)=-\frac{3}{2}\sqrt{\mu}a^{-\frac{5}{2}}\Delta at. \tag{3}\]  
where \(\Delta a\) is the amount of attenuation caused by atmospheric drag.  
According to the phase calculation formula of the reference star,  
\[\Delta\lambda=\Delta\lambda(t)+\Delta\dot{\lambda}. \tag{4}\]  
When the phase deviation \(\Delta\lambda\) of the satellite relative to the reference star exceeds the threshold value, the following condition is satisfied:  
\[|\Delta\lambda|>\lambda_{\text{max}}. \tag{5}\]  
The orbit control of the satellite needs to be implemented.