Frozen Orbits

Due to the Earth's gravitational perturbation, the orbit of a salite will experience secular changes in the argument of perigee. Hence, the satellite mean altitude per latitude will differ during the mission. This effect can be problematic, especially if we must compare images by a camera onboard the satellite in different periods. The altitude variation will change the resolution, leading to some problems when comparing the data.

We can avoid this problem if we compute an eccentricity $e$ and the argument of perigee $\omega$ that yields theoretically:

\[\begin{equation*} \frac{de}{dt} = 0,\ \frac{d\omega}{dt} = 0\ . \end{equation*}\]

This orbit is called frozen. Refer to [1] for more information.

We can use the function:

frozen_orbit(a::Number, i::Number; kwargs...) -> Float64, Float64

to compute the eccentricity [ ] and argument of perigee [rad] that yield a frozen orbit when the orbit has semi-major axis a [m] and inclination i [rad]. This function uses the theory in [1].

The following keywords are available:

• gravity_model::Union{Nothing, AbstractGravityModel}: Gravity model used to compute the frozen eccentricity. Refer to the object AbstractGravityModel of the package SatelliteToolboxGravityModels.jl for more information. If it is nothing, the system will automatically fetch and load the EGM96 gravity model. However, loading a gravity model can significantly decrease the performance. Thus, it is advisable to pass a gravity model here. (Default = nothing)
• max_degree: Maximum gravity model degree used to compute the frozen eccentricity. If it is equal to or lower than 0, the maximum degree in grav_model will be used. Otherwise, if it is lower than 3 or higher than the grav_model maximum degree, it will be clamped accordingly. (Default = 53)

Examples

We will compute the eccentricity and argument of perigee that yields a frozen orbit using the data from Amazonia-1 mission. First, we will use only 5 degrees, and the default gravity model (EGM96):

julia> frozen_orbit(7130.982e3, 98.410 |> deg2rad; max_degree = 5)(0.0011108978494835141, 1.5707963267948966)
julia> e, ω = frozen_orbit(7130.982e3, 98.410 |> deg2rad; max_degree = 5)(0.0011108978494835141, 1.5707963267948966)
julia> e0.0011108978494835141
julia> ω |> rad2deg90.0

If we want to use all the 360 terms in EGM96, we need to increase the precision of BigFloat to keep the accuracy:

julia> setprecision(1024)1024
julia> e, ω = frozen_orbit(7130.982e3, 98.410 |> deg2rad; max_degree = 5)(0.0011108978494835141, 1.5707963267948966)
julia> e0.0011108978494835141
julia> ω |> rad2deg90.0

We can use a different gravity model as follows:

julia> jgm3 = GravityModels.load(IcgemFile, fetch_icgem_file(:JGM3))IcgemFile{Float64}:
      Product type : gravity_field
       Model name  : JGM3
  Gravity constant : 3.986004415e14
            Radius : 6.3781363e6
    Maximum degree : 70
            Errors : formal
       Tide system : unknown
              Norm : fully_normalized
         Data type : Float64
julia> e, ω = frozen_orbit(7130.982e3, 98.410 |> deg2rad; max_degree = 70, gravity_model = jgm3)(0.001163504566870769, 1.5707963267948966)
julia> e0.001163504566870769
julia> ω |> rad2deg90.0

References

• [1] Rosborough, G. W.; Ocampo, C. A (1991). Influence of higher degree zonals on the frozen orbit geometry. Proceedings of the AAS/AIAA Astrodynamics Conference, Durango, CO.