Minimum Space Occupancy (MiSO) Orbits

• Minimum Space Occupancy (MiSO) orbits are satellite trajectories that minimize the three-dimensional space occupied over time under full perturbative effects.
• They incorporate all key perturbations—zonal, tesseral, lunisolar, SRP, and drag—via iterative numerical optimization to reduce metrics like space-occupancy range, area, and volume.
• MiSO concepts enable non-intersecting, slot-based orbital architectures that drastically lower conjunction events and boost overall mega-constellation capacity.

Minimum Space Occupancy (MiSO) orbits are a class of satellite trajectories in which the three-dimensional domain traversed by a spacecraft over a finite time interval is minimized, subject to the full suite of Earth-based and external perturbations. The concept, initially formulated in the context of mega-constellations and space traffic management, generalizes the classical notion of frozen orbits to fully perturbed, non-averaged dynamics. MiSO orbits are specified by osculating initial conditions that minimize metrics such as the space-occupancy range (SOR), area (SOA), or volume (SOV), thus reducing the spatial overlap of satellite “tubes” and drastically mitigating intra-constellation conjunctions and collision risk (Bombardelli et al., 2020, Reiland et al., 2020, Arnas et al., 2021).

1. Definitions and Theoretical Foundations

The space-occupancy of a satellite is quantified via three nested measures:

• Space-Occupancy Range (SOR): The maximum altitude variation $$\mathrm{SOR}(t_0, \Delta t) = \max_{\phi, t} \Delta r(\phi, t)$$ at fixed geocentric latitude $\phi$ over a prescribed timespan $[t_0, t_0+\Delta t]$, with $\phi_{\max} \approx i$ (mean inclination).
• Space-Occupancy Area (SOA): In the mean orbital plane, the minimal area that covers the trajectory during the interval, simplifying to an annulus of thickness $\mathrm{SOR}$ for pure-zonal (J$_2$-only) perturbations.
• Space-Occupancy Volume (SOV): The 3D volume swept out as the SOA precesses around Earth's spin axis.

For singly-averaged $J_2$ dynamics, the SOR is directly related to the proper eccentricity $e_p$: $\mathrm{SOR} \approx 2 \hat{a} e_p$. Minimizing $e_p$ (and thus SOR) under the full perturbation model yields a MiSO orbit. Given the density and almost coplanarity of mega-constellations, minimizing SOR, SOA, and SOV directly reduces critical conjunctions and collision-avoidance complexity (Bombardelli et al., 2020).

2. From Frozen Orbits to MiSO Orbits: Generalization and Formulation

Classical frozen orbits are secular equilibria of the $J_2$ Hamiltonian, yielding approximately constant eccentricity and argument of perigee via $$\partial \bar{\mathcal{R}}_{J_2} / \partial \omega = 0$$ ($\omega = 90^\circ/270^\circ$) and $\partial \bar{\mathcal{R}}_{J_2} / \partial e = 0$, most notably at the critical inclinations near $63.4^\circ$ (Reiland et al., 2020). MiSO orbits generalize this to the fully perturbed system—including tesseral harmonics, lunisolar third-body terms, solar radiation pressure (SRP), and atmospheric drag—by seeking osculating elements $x = [a_0, e_0, i_0, \Omega_0, \omega_0, M_0]$ that minimize the true, non-averaged spatial occupancy (Bombardelli et al., 2020, Reiland et al., 2020).

An explicit optimization is performed over $$J(x) = \max_{t, \phi} r(x, t; \phi) - \min_{t, \phi} r(x, t; \phi)$$, often over $100$-day windows and subject to mission constraints (perigee, inclination, safety margins). The choice of propagator is critical; studies employ non-averaged, regularized integrators such as THALASSA (using Dromo or EDromo regularizations) to accurately capture all perturbations (Reiland et al., 2020, Bombardelli et al., 2020).

3. Non-Self-Intersecting Trajectories and Slotting Algorithms

In the context of collision-free constellation architectures, MiSO orbits are directly connected to the theory of non-self-intersecting relative trajectories in a rotating (terrestrial-fixed) frame. The central result is that, for circular orbits, non-self-intersection is guaranteed if and only if $|N_p - N_d| = 1$, where $N_p, N_d$ are coprime integers relating mean motion $n$ to Earth spin rate $\alpha$ via $\alpha/n = \pm N_d/N_p$ (Arnas et al., 2021). For a fixed $(a, i)$, Theorem 2 in (Arnas et al., 2021) provides necessary and sufficient inequalities on inclination and $(N_p, N_d)$ to construct a non-self-intersecting loop in the $(\Delta \Omega, \Delta M)$ plane.

This underpinning allows the construction of multi-satellite slotting architectures (the MiSO recipe):

1. Select $(N_p, N_d)$ per Theorem 2 for the given inclination.
2. Compute along-track slot count $$N_s \approx \left\lfloor \frac{2\pi}{\rho_{\min} (N_p \mp N_d \cos i)} \right\rfloor$$ for given separation $\rho_{\min}$.
3. Compute cross-track plane count $$M \approx \left\lfloor \frac{2\pi a |\sin i|}{\rho_{\min}} \right\rfloor$$.
4. Assign positions in $(\Omega, M)$ using explicit phasing for minimal pairwise distance.
5. Verify minimal separation over $[0, T_c)$ using closed formulas for normalized distance (Arnas et al., 2021).

Total shell capacity for a spherical shell at altitude $a$ is thus $$N_{\text{total}} \approx N_s \times M \propto 1/\rho_{\min}^2$$ for small $\rho_{\min}$. Robustness to small parameter drifts is inherent as conjunction avoidance is preserved unless the minimal spacing is violated.

4. Numerical Optimization and Implementation

MiSO initial conditions are derived by iterative numerical optimization, starting from frozen-orbit guesses, with the following typical workflow (Bombardelli et al., 2020, Reiland et al., 2020):

• Initialization: Set $(a_0, e_0, \omega_0)$ to frozen-orbit values: $e_f \simeq \frac{3J_3}{2J_2} \sin \hat{i}$, $\omega_0\simeq 90^\circ$.
• Integration: Propagate the full, non-averaged equations over $\Delta t$ (e.g., 100 days).
• SOR Evaluation: At each time/latitude, record $r(x,t;\phi)$, binned in $\phi$, to compute the SOR.
• Optimization: Use a derivative-free or gradient-based algorithm to minimize SOR (e.g., grid search, Nelder–Mead), refining $x$ iteratively with convergence criteria of 10 cm or better.
• Constraints: Impose perigee altitude, eccentricity, and ground-track constraints as required.

This delivers osculating vectors $x$ for seed satellites, with plane populations obtained by subsampling trajectory phases.

Representative Results

Orbit Class	Plane	Frozen SOR [m]	MiSO SOR [m]	Reduction [m]
h$_N$=550 km	0°	575	503	72
h$_N$=550 km	90°	754	497	257
h$_N$=1168 km	0°	460	378	82
h$_N$=1168 km	120°	560	393	167

In the presence of drag and SRP, MiSO still reduces SOR by 100–400 m compared to frozen orbits, with Δ$V$ requirements well within standard insertion and station-keeping tolerances (Bombardelli et al., 2020).

5. Applications to Mega-Constellation and Orbital Capacity

MiSO architectures enable stacking of non-overlapping shells in altitude, with each shell separated by the SOR of the lower shell: $h_{N,p+1}=h_{N,p}+\mathrm{SOR}_p$ (Bombardelli et al., 2020). The direct outcome is the virtual elimination of inter-plane conjunctions, reducing collision-avoidance maneuvers dramatically. For example, a OneWeb-like LEO deployment (36 planes, 55 satellites per plane) exhibits a reduction from 2522 to 232 close approaches (within 1 km) over 90 days upon adopting MiSO: a $\sim$91% reduction. Similar reductions are obtained for representative Starlink shells (Reiland et al., 2020).

Closed-form expressions in the MiSO slotting framework permit analytical estimation of shell capacity for arbitrary altitude, inclination, and safety spacing, optimizing network design and minimizing spectrum and space resource conflicts (Arnas et al., 2021). Architectures constructed according to the MiSO recipe are provably non-conjunctionable for satellites conforming to nominal parameter bounds.

6. Trade-offs, Sensitivity, and Design Guidelines

Key considerations in MiSO design include:

• Inclination-induced limits: Maximum $N_p$ (number of loop closures per cycle) dictated by inclination; higher $N_p$ slightly decreases along-track capacity.
• Separation requirements: SOR and thus shell separation set by mission-specific safety margins.
• Robustness: Cross- and along-track spacing remains resilient to small parameter drifts ($\lesssim 0.1^\circ$ in $\omega_0$ due to tesseral order changes).
• Station-keeping: MiSO minimizes need for frequent maneuvers; only slow secular drift in $\Omega$ (nodal regression) and $\omega$ due to lunisolar effects require routine Δ$V$ at multi-month intervals.
• Total capacity scaling: $N_{\text{total}}\propto 1/\rho_{\min}^2$ (for small $\rho_{\min}$), with cross- and along-track densities tunable independently, allowing flexible response to regulatory and coverage constraints.

An explicit trade-off table is provided in (Arnas et al., 2021) for typical LEO settings, relating cycle length $T_c$, $N_p$, $N_s$, $M$, and total capacity.

7. Broader Implications and Future Directions

MiSO orbits supply a rigorous, perturbation-inclusive generalization of frozen orbits, yielding practical recipes for the design of conjunction-free mega-constellations and scalable slotting architectures (Bombardelli et al., 2020, Arnas et al., 2021). The MiSO approach directly answers the impending need for formal congestion management and orbital capacity regulation in the era of proliferating low-Earth-orbit assets.

A plausible implication is that MiSO architectures—through their explicit avoidance of self-intersections and minimal occupancy “tubes”—can inform global slotting standards, spectrum allocation, and space traffic management protocols. Generalization to higher eccentricity regimes and dynamical extensions remain areas for further research, as does integration with operational station-keeping and reconfiguration strategies.