Determining the Keplerian Elements of an Elliptical Orbit from Four Observations
Presented hereafter is a method for determining a preliminary heliocentric orbit from four geocentric directions of a sun-orbiting object at four distinct times of observation. I take this method after that presented in chapter six of The Determination of Orbits by A.D. Dubyago.
The initial data
t₁, X⊕₁, Y⊕₁, Z⊕₁, α₁, δ₁
t₂, X⊕₂, Y⊕₂, Z⊕₂, α₂, δ₂
t₃, X⊕₃, Y⊕₃, Z⊕₃, α₃, δ₃
t₄, X⊕₄, Y⊕₄, Z⊕₄, α₄, δ₄
The times of observation, tᵢ, are given in Julian date format. The vectors [X⊕ᵢ,Y⊕ᵢ,Z⊕ᵢ] are t