## 4 Conclusions  
This paper provided an in-depth study on the optimal distribution of satellite constellations and offers a theoretical framework applicable to clusters of arbitrary numbers of satellites. For Earth scanning missions with circular orbits and target pointing attitude, this study developed a mathematical model to optimize the relative distribution of an arbitrary number of satellites equipped with receivers capable of performing AOA measurements.  
Extensive simulations were performed for systems composed of two satellites with the purpose of geolocating one signal emitter located on the surface of Earth. In particular, quantitative results were presented in terms of differences in the angles of the right ascension of the ascending node \(\Delta\Omega\), differences in the angles of the true anomaly \(\Delta\
u\), and the spacing angle \(\gamma_{1,2}\) between the satellites with respect to the emitter position. Qualitative trends were also discussed, which fully confirm the previously predicted properties of symmetry and anti-symmetry for the difference in the right ascension of the ascending node \(\Delta\Omega\) and the difference in true anomalies \(\Delta\
u\) and partially confirm symmetric properties for spacing angle \(\gamma\) obtained in a previous study by the authors [26] while taking into account variation of the configurations in terms of orbit radius \(r\) and the position of the emitter in terms of latitude \(\Lambda\).  
This research provides a guide for translating mission planning requirements into satellite constellation designs for the operative case of the placement of a second companion satellite after a first one is launched.  
The theoretical analysis of the trends of two-satellite systems presented in this paper can also support the building of data-driven fitting and modeling for simulators, facilitating fast predictions of constellation designs, at least for two satellites, and configuration comparisons, providing beneficial insights for decision-makers in the space domain.  
It is noted that among the limitations of this study to be addressed in future activities are the influence of the attitude of the satellites and the effect due to the signal-to-noise ratio. Also, third party validation of the results and comparison with other optimization approaches or strategies to model the constraints could be explored in future works. Furthermore, future studies could expand on this work by simulating constellations with more satellites, incorporating multiple measurements over time, and exploring more complex scenarios by relaxing some of the assumptions made in this study.  
**Author Contributions:** Conceptualization, investigation: [PERSON], [PERSON]-[PERSON], and [PERSON] ideated the general approach and potential applications of this study. Methodology, software, writing (original draft): [PERSON] developed the mathematical modeling, performed the numerical simulations, and wrote the original draft of this paper. Validation, writing (review and editing): [PERSON], [PERSON], [PERSON] and [PERSON] performed the operations of verifying, reviewing, and editing this paper. Supervision: [PERSON] and [PERSON] supervised the project. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Dataset available upon request.  
**Conflicts of Interest:** The authors declare no conflicts of interest.