# A Study of Cislunar-Based Small Satellite Constellations with Sustainable Autonomy
## 4 Cislunar Dynamics (Constellation Scenario)
### Cislunar Debris  
The growing concerns with Cislunar debris have to be addressed through dedicated dynamic modeling and understanding of the possible debris model in the region. Various parameters have been studied and analyzed to be incorporated in the simulational scenarios on Earth and Moon constellations, respectively. Hence, the equations mentioned in [33] have been significantly useful in aligning the debris model as given below:  
Debris model:  
Given the kinetic energy of a particle is considered constant, then the particle speed is calculated as:  
\[\partial=\sqrt{\frac{2}{m}}\kappa \tag{14}\]  
where \(\kappa\) is the kinetic energy, and \(m\) is the particle mass.  
The probability of the critical damage given an impact happening in orbit is determined by a vulnerability model based on the properties of the debris in the surroundings of the small satellite. Given this model, the probability of the damage is given as:  
\[P_{D}=1-e^{-E\ast P_{K|H}} \tag{15}\]  
\[E=\zeta\ast V_{k} \tag{16}\]  
where \(P_{K|H}\) is the instantaneous probability of spacecraft hazard, \(\zeta\) is the debris number density, and \(V_{k}\) is the hazard zone volume.  
When the total time of the damage from start to end is considered, the following expression is aligned: This expression redefines the difference in the initial timing of the disaster occurrence to the final time the impact was observed. The integral form of this variation can be observed with the Equation (17), which is the most dynamic part of operations.  
\[\int\limits_{t_{0}}^{t_{f}}P_{D}(t)dt \tag{17}\]  
Referring back to [26], it mentions the surveillance factor for the selected orbits in the Cislunar space. This is to develop an optical sensor to monitor the vicinity of the spacecraftwhen needed and adjust its proximity as needed. The magnitude of a space object in the vicinity of this satellite is framed in an equation as:  
\[\psi=\Psi_{sun}-2.5\log_{10}\left(\frac{I_{S}}{I_{Sun}}\right) \tag{18}\]  
where \(\Psi_{sun}\) is the apparent reference magnitude of the Sun and \(I_{S}\) is the irradiance reflected from the spacecraft and Sun's reference irradiance.  
Moving forward, this paper will add a unique dimension to the above-mentioned equations and debris-determining techniques by aligning a dedicated Monte Carlo simulation estimating the close approaches to each spacecraft in constellation using a well-designed Gaussian distribution over thousands of states of a satellite in a given scenario emphasizing the closest and farthest probability of collision. Hence, a detailed study to align the equations for the execution of this space situational awareness task with the proposed constellation designs and also during the transit scenario from Earth to the Moon is duly considered and analyzed. The equations for the Gaussian distribution are referred to from [34] as:  
If \(X\) is the initial state of the satellite and has a Gaussian distribution with a mean \(\mu\) and covariance matrix as \(\Sigma\), then the distribution of the \(X\) for \(n\) trials is given by:  
\[f_{x}(X)=\frac{1}{\sqrt{\left(2\Pi\right)^{n}\left|\Sigma\right|}}\exp\Bigl{\{} -0.5(X-\mu)^{T}\Sigma^{-1}(X-\mu)\Bigr{\}} \tag{19}\]  
The total derivation and explanation are given in [34] significantly contributing towards SSA and its related uncertainties. This will add essence with thorough understanding and firm establishing of the work. The basis of this Monte Carlo analysis is to calculate and evaluate the number of close misses with a resulting probability of collision for thousands of random states, with the close approach close misses and grouped points, and with time of approach in a span of propagation and within a probable area of collision.