Two Body Analytical Propagator

The two-body analytical orbit propagator considers the Earth a perfect sphere with uniform density. Hence, it propagates the orbit using the solution considering Newtonian gravity. It has an extremely low precision but with a minimal computational burden. Thus, it is helpful in some analysis that requires propagating the orbit many times for short periods.

Algorithm

The algorithm implemented here is based on [1].

Since we are considering a spherical Earth with uniform density, gravity points towards the center of Earth. Thus, we propagate the orbit by updating the satellite mean anomaly since all other Keplerian elements do not change. The equation to correct the mean anomaly is:

\[M(t) = M_0 + \sqrt{\frac{\mu}{a_0^3}} \cdot \left(t - t_0\right)\]

where $t_0$ is the initial mean elements' epoch, $a_0$ is the mean semi-major axis, and $\mu$ the Earth's standard gravitational parameter.

Initialization

We can initialize the two-body analytical propagator with the following function:

Propagators.init(Val(:TwoBody), orb₀::KeplerianElements; kwargs...) -> OrbitPropagatorTwoBody

which creates a two-body propagator structure OrbitPropagatorTwoBody with the mean Keplerian elements orb₀. The following keyword selects the standard gravitational parameter for the propagation algorithm:

m0::T: Standard gravitational parameter of the central body [m³/s²]. (Default = tbc_m0)

This package contains some pre-built gravitational parameters of the Earth for this propagator:

Two-Body Propagator Constant	Description	Type
`tbc_m0`	Earth's standard gravitational parameter	`Float64`
`tbc_m0_f32`	Earth's standard gravitational parameter	`Float32`

Note

The type used in the propagation will be the same as used to define the gravitational constant μ.

julia> orb = KeplerianElements(
                  date_to_jd(2023, 1, 1, 0, 0, 0),
                  7190.982e3,
                  0.001111,
                  98.405 |> deg2rad,
                  100    |> deg2rad,
                  90     |> deg2rad,
                  19     |> deg2rad
              )KeplerianElements{Float64, Float64}:
           Epoch :    2.45995e6 (2023-01-01T00:00:00)
 Semi-major axis : 7190.98     km
    Eccentricity :    0.001111
     Inclination :   98.405    °
            RAAN :  100.0      °
 Arg. of Perigee :   90.0      °
    True Anomaly :   19.0      °
julia> orbp = Propagators.init(Val(:TwoBody), orb)OrbitPropagatorTwoBody{Float64, Float64}:
   Propagator name : Two-Body Orbit Propagator
  Propagator epoch : 2023-01-01T00:00:00
  Last propagation : 2023-01-01T00:00:00

References

[1] Vallado, D. A (2013). Fundamentals of Astrodynamics and Applications. 4th ed. Microcosm Press, Hawthorn, CA, USA.