Transformations of Orbit Representations

Orbit State Vector

We provide a set of functions to transform an OrbitStateVector between any frame described in Transformations Between ECEF and ECI Reference Frames.

From ECI to ECI

The function

sv_eci_to_eci(sv::OrbitStateVector, args...) -> OrbitStateVector

can be used to transform the OrbitStateVector sv from one ECI frame to another. The arguments args... must match those of the function r_eci_to_eci without the rotation representation.

The following example shows how we can convert a state vector from the MOD (Mean of Date) reference frame to the TOD (True of Date) reference frame:

julia> jd_utc = date_to_jd(2004, 4, 6, 7, 51, 28.386009)2.453101827411875e6
julia> r_mod  = [5094.02837450; 6127.87081640; 6380.24851640]3-element Vector{Float64}:
 5094.0283745
 6127.8708164
 6380.2485164
julia> v_mod  = [-4.7462630520; 0.7860140450; 5.5317905620]3-element Vector{Float64}:
 -4.746263052
  0.786014045
  5.531790562
julia> sv_mod = OrbitStateVector(jd_utc, r_mod, v_mod)OrbitStateVector{Float64, Float64}:
  epoch : 2.4531e6 (2004-04-06T07:51:28.386)
      r : [5.09403, 6.12787, 6.38025]             km
      v : [-0.00474626, 0.000786014, 0.00553179]  km/s
julia> sv_tod = sv_eci_to_eci(sv_mod, MOD(), jd_utc, TOD(), jd_utc)OrbitStateVector{Float64, Float64}:
  epoch : 2.4531e6 (2004-04-06T07:51:28.386)
      r : [5.09451, 6.12737, 6.38035]             km
      v : [-0.00474609, 0.000786077, 0.00553193]  km/s

From ECI to ECEF

The function

sv_eci_to_ecef(sv::OrbitStateVector, ECI, ECEF, jd_utc[, eop]) -> OrbitStateVector

can be used to convert the orbit state vector sv from the Earth-Centered Inertial (ECI) reference frame ECI to the Earth-Centered, Earth-Fixed (ECEF) reference frame at the Julian day jd_utc [UTC]. The eop may be required depending on the selection of the input and output reference system. For more information, see the documentation of the function r_eci_to_ecef.

Info

It is assumed that the input velocity and acceleration in sv are obtained by an observer on the ECI frame. Thus, the output will contain the velocity and acceleration as measured by an observer on the ECEF frame.

The following example shows how we can convert a state vector from the J2000 reference frame reference frame to PEF (True of Date) reference frame:

julia> jd_ut1 = date_to_jd(2004, 4, 6, 7, 51, 28.386009) - 0.4399619 / 864002.453101827406783e6
julia> r_j2000  = [5102.50960000; 6123.01152000; 6378.13630000]3-element Vector{Float64}:
 5102.5096
 6123.01152
 6378.1363
julia> v_j2000  = [-4.7432196000; 0.7905366000; 5.5337561900]3-element Vector{Float64}:
 -4.7432196
  0.7905366
  5.53375619
julia> sv_j2000 = OrbitStateVector(jd_ut1, r_j2000, v_j2000)OrbitStateVector{Float64, Float64}:
  epoch : 2.4531e6 (2004-04-06T07:51:27.946)
      r : [5.10251, 6.12301, 6.37814]             km
      v : [-0.00474322, 0.000790537, 0.00553376]  km/s
julia> sv_pef   = sv_eci_to_ecef(sv_j2000, J2000(), PEF(), jd_ut1)OrbitStateVector{Float64, Float64}:
  epoch : 2.4531e6 (2004-04-06T07:51:27.946)
      r : [-1.03348, 7.90131, 6.38034]            km
      v : [-0.00322563, -0.00287244, 0.00553193]  km/s

From ECEF to ECI

The function

sv_ecef_to_eci(sv::OrbitStateVector, ECEF, ECI, jd_utc[, eop]) -> OrbitStateVector

can be used to convert the orbit state vector sv from the Earth-Centered, Earth-Fixed (ECEF) reference frame ECEF to the Earth-Centered Inertial (ECI) reference frame at the Julian day jd_utc [UTC]. The eop may be required depending on the selection of the input and output reference system. For more information, see the documentation of the function r_ecef_to_eci.

Info

It is assumed that the input velocity and acceleration in sv are obtained by an observer on the ECEF frame. Thus, the output will contain the velocity and acceleration as measured by an observer on the ECI frame.

The following example shows how we can convert a state vector from the PEF reference frame reference frame to J2000 reference frame:

julia> jd_ut1 = date_to_jd(2004, 4, 6, 7, 51, 28.386009) - 0.4399619 / 864002.453101827406783e6
julia> r_pef = [-1033.47503130; 7901.30558560; 6380.34453270]3-element Vector{Float64}:
 -1033.4750313
  7901.3055856
  6380.3445327
julia> v_pef = [-3.2256327470; -2.8724425110; +5.5319312880]3-element Vector{Float64}:
 -3.225632747
 -2.872442511
  5.531931288
julia> sv_pef   = OrbitStateVector(jd_ut1, r_pef, v_pef)OrbitStateVector{Float64, Float64}:
  epoch : 2.4531e6 (2004-04-06T07:51:27.946)
      r : [-1.03348, 7.90131, 6.38034]            km
      v : [-0.00322563, -0.00287244, 0.00553193]  km/s
julia> sv_j2000 = sv_ecef_to_eci(sv_pef, PEF(), J2000(), jd_ut1)OrbitStateVector{Float64, Float64}:
  epoch : 2.4531e6 (2004-04-06T07:51:27.946)
      r : [5.10251, 6.12301, 6.37814]             km
      v : [-0.00474322, 0.000790537, 0.00553376]  km/s

From ECEF to ECEF

The function

sv_ecef_to_ecef(sv::OrbitStateVector, args...) -> OrbitStateVector

can be used to transform the orbit state vector sv from an ECEF frame to another ECEF frame. The arguments args... must match those of the function r_ecef_to_ecef wihtout the rotation representation.

The following example shows how we can convert a state vector from the ITRF reference frame reference frame to PEF reference frame:

julia> eop_iau1980 = fetch_iers_eop()EopIau1980:
     Data │ Timespan
 ─────────┼──────────────────────────────────────────────
        x │ 1973-01-02T00:00:00 -- 2025-05-31T00:00:00
        y │ 1973-01-02T00:00:00 -- 2025-05-31T00:00:00
  UT1-UTC │ 1973-01-02T00:00:00 -- 2025-05-31T00:00:00
      LOD │ 1973-01-02T00:00:00 -- 2024-05-22T00:00:00
      δΔψ │ 1973-01-02T00:00:00 -- 2024-08-05T00:00:00
      δΔϵ │ 1973-01-02T00:00:00 -- 2024-08-05T00:00:00
julia> jd_utc = date_to_jd(2004, 4, 6, 7, 51, 28.386009)2.453101827411875e6
julia> r_itrf = [-1033.4793830; 7901.2952754; 6380.3565958]3-element Vector{Float64}:
 -1033.479383
  7901.2952754
  6380.3565958
julia> v_itrf = [-3.225636520; -2.872451450; +5.531924446]3-element Vector{Float64}:
 -3.22563652
 -2.87245145
  5.531924446
julia> sv_itrf = OrbitStateVector(jd_utc, r_itrf, v_itrf)OrbitStateVector{Float64, Float64}:
  epoch : 2.4531e6 (2004-04-06T07:51:28.386)
      r : [-1.03348, 7.9013, 6.38036]             km
      v : [-0.00322564, -0.00287245, 0.00553192]  km/s
julia> sv_pef = sv_ecef_to_ecef(sv_itrf, ITRF(), PEF(), jd_utc, eop_iau1980)OrbitStateVector{Float64, Float64}:
  epoch : 2.4531e6 (2004-04-06T07:51:28.386)
      r : [-1.03348, 7.90131, 6.38034]            km
      v : [-0.00322563, -0.00287244, 0.00553193]  km/s