### Numerical atlas: GEO survey

This section contains the results from the exhaustive dynamical study around the geosynchronous altitude conducted in the

framework of the H2020 ReDHIFT project in Politecnico di Milano [1],[2].

A few hundred million orbits were propagated with the single-averaged PlanODyn suite [3], using a 4×4 geopotential, lunisolar perturbations (up to 5th order in the parallax factor), a cannonball solar radiation pressure model and taking also into account the effects of Earth’s precession.

Three particular configurations were selected to present the results for maximum clarity:

framework of the H2020 ReDHIFT project in Politecnico di Milano [1],[2].

A few hundred million orbits were propagated with the single-averaged PlanODyn suite [3], using a 4×4 geopotential, lunisolar perturbations (up to 5th order in the parallax factor), a cannonball solar radiation pressure model and taking also into account the effects of Earth’s precession.

Three particular configurations were selected to present the results for maximum clarity:

- The “eye” maps
- The “Omega-omega” maps
- The “angles averaged” maps

### Eye maps

The “eye” maps were designed to explore the contribution of the tesseral effects on the eccentricity growth for geosynchronous orbits.

The main grid is over the initial satellite semi-major axis (a) and the initial resonant angle for the geosynchronous resonance (λ = Ω+ω+M – θ g ). The grid in the semi-major axis ranges from 100 km below to 100 km above the geosynchronous value (a = 42164 km) and the resonant angle λ from 0 to 360 degrees.

The computation of the grid is repeated for a selected set of the remaining initial orbit parameters, eccentricity and inclination, and for 2 values of area to mass ratio 0.012 m² /kg, and 1.0 m² /kg.

The maximum variation in the eccentricity is presented with a color code in each map.

The main grid is over the initial satellite semi-major axis (a) and the initial resonant angle for the geosynchronous resonance (λ = Ω+ω+M – θ g ). The grid in the semi-major axis ranges from 100 km below to 100 km above the geosynchronous value (a = 42164 km) and the resonant angle λ from 0 to 360 degrees.

The computation of the grid is repeated for a selected set of the remaining initial orbit parameters, eccentricity and inclination, and for 2 values of area to mass ratio 0.012 m² /kg, and 1.0 m² /kg.

The maximum variation in the eccentricity is presented with a color code in each map.

### Ω − ω maps

The “Omega-omega” maps were designed to explore the contribution of lunisolar perturbations to distant satellite orbits and their interaction with solar radiation pressure effects.

The main grid is over the initial longitude of the ascending node and the initial argument of the perigee, the two angles that appear in the lunisolar and solar radiation pressure single-averaged perturbations. Although the third body perturbations are symmetrical over π with respect to the argument of the perigee, both angles in the grid vary from 0 to 360 degrees, because the SRP effects break that symmetry.

The computation of the grid is repeated for a selected set of the other initial orbit parameters (a,e,i), and for 2 values of area to mass ratio 0.012 m 2 /kg, and 1.0 m 2 /kg. The maximum variation in the eccentricity is presented with a color code in each map.

The main grid is over the initial longitude of the ascending node and the initial argument of the perigee, the two angles that appear in the lunisolar and solar radiation pressure single-averaged perturbations. Although the third body perturbations are symmetrical over π with respect to the argument of the perigee, both angles in the grid vary from 0 to 360 degrees, because the SRP effects break that symmetry.

The computation of the grid is repeated for a selected set of the other initial orbit parameters (a,e,i), and for 2 values of area to mass ratio 0.012 m 2 /kg, and 1.0 m 2 /kg. The maximum variation in the eccentricity is presented with a color code in each map.

Low Area-to-Mass ratio

High Area-to-Mass ratio

### Angles averaged maps

The “averaged over the angles” maps were designed to globally scan the (a, e, i) space for regions were good graveyard or good re-entry solutions exist.

The grids are set up by fixing one of the parameters the semi major-axis and computing several maps in the e − i plane for various randomly selected sets of initial angles (ω,Ω and M). The maximum averaged variation of the eccentricity is presented with a color code in each map.

The grids are set up by fixing one of the parameters the semi major-axis and computing several maps in the e − i plane for various randomly selected sets of initial angles (ω,Ω and M). The maximum averaged variation of the eccentricity is presented with a color code in each map.

### Contacts

Department of Aerospace Science and Technology, Politecnico di Milano, Italy

- Camilla Colombo camilla.colombo@polimi.it
- Ioannis Gkolias ioannis.gkolias@polimi.it